sequences and series - · pdf fileproblem solving crf, every lesson ... holt authors ed burger...

Lesson/Lab California Standards Time

AdvancedStudents

Benchmark*Students

Strategic**Students

- ntrodu tion to Se uen es Prep for 22.0 50 min

N eometri atterns and essellations

Prep for 22.0 50 min

- Series and Summation Notation

Prep for 23.0 50 min

L valuate Se uen es and Series

Prep for 22.0, Prep for 23.0 50 min

-3 rit meti Se uen es and Series

22.0, 23.0 50 min

on e t onne tion 10 min

eady to o n 15 min

-4 eometri Se uen es and Series

22.0, 23.0 75 min

L lore nfi nite eometri Series

23.0 50 min

-5 at emati al ndu tion and nfi nite eometri Series

21.0, 22.0, 23.0 75 min

X rea Under a urve Ext of 22.0 50 min

on e t onne tion 10 min

eady to o n 15 min

Study uide: evie

a ter est 50 min

* Benchmark students are achieving at or near grade level. ** Strategic students may be a year or more below grade-level,

and may require additional time for intervention.

Pacing Guide

RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE

Sequences and Series

858A Chapter 12

Grade-level Standard

ReviewBeyond the StandardsAssessment

Optional

E OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS

858BKEY: SE = Student Edition TE = Teacher’s Edition CRF = Chapter Resource File AR = Assessment Resources Available online Available on CD-ROM

SummativeAssessment

After Chapter 12Check mastery of chapter concepts.

Multiple-Choice Tests (Forms A, B, C)Free-Response Tests (Forms A, B, C)Performance Assessment AR pp. 239–240

Test and Practice Generator

Prescribe intervention.

Review for Mastery CRF, every lesson

Lesson Tutorial Videos Chapter 12

Before Chapter 12 TestingDiagnose mastery of concepts in the chapter.

Ready to Go On? SE pp. 889, 909

Concept Connection SE pp. 888, 908

Section Quizzes AR pp. 225–226



Ready to Go On? Intervention Chapter 12

Scaffolding Questions TE pp. 888, 908

Before Assessment of California StandardsDiagnose mastery of California Standards.

Focus on CAHSEE Standards: Benchmark TestsStandardized Test Prep SE pp. 920–921

California Standards Practice CD-ROMCAHSEE Standards Practice CD-ROM


Focus on CAHSEE Standards: Intervention

Before Every LessonDiagnose readiness for the lesson.

Warm Up TE, every lesson


Skills Bank pp. S46–S73

Review for Mastery CRF Chapters 1–12

During Every LessonDiagnose understanding of lesson concepts.

Questioning Strategies TE, every example

Check It Out! SE, every example

Think and Discuss SE, every lesson

Write About It SE, every lesson

Journal TE, every lesson


Reading Strategies CRF, every lesson

Success for ELL pp. 167–176

Lesson Tutorial Videos Chapter 12

After Every LessonDiagnose mastery of lesson concepts.

Lesson Quiz TE, every lesson

Alternative Assessment TE, every lesson

Ready to Go On? SE pp. 889, 909



Review for Mastery CRF, every lesson

Problem Solving CRF, every lesson

Ready to Go On? Intervention Chapter 12

Homework Help Online

DIAGNOSE PRESCRIBE

AssessPrior

Knowledge

Before Chapter 12Diagnose readiness for the chapter.

Are You Ready? SE p. 859


Are You Ready? Intervention Skills 8, 53, 60, 72, 89

FormativeAssessment


858C Chapter 12

Power Presentations One-Stop Planner Premier Online Edition Dynamic presentations to engage students. Complete PowerPoint® presentations for every lesson in Chapter 12.

Easy access to Chapter 12 resources and assessments. Includes lesson planning, test generation, and puzzle creation software.

Includes Tutorial Videos, Lesson Activities, Lesson Quizzes, Homework Help, Chapter Project and more.

Supporting the Teacher

Technology Highlights for the Teacher

KEY: SE = Student Edition TE = Teacher’s Edition English Language Learners Spanish available Available online Available on CD-ROM

Chapter 12 Resource FileParent Letter

pp. 1–2

Practice A, B, C pp. 3–5, 11–13, 19–21, 27–29, 35–37

Review for Mastery pp. 6–7, 14–15, 22–23, 30–31, 38–39

Challenge pp. 8, 16, 24, 32, 40

Problem Solving pp. 9, 17, 25, 33, 41

Reading Strategies pp. 10, 18, 26, 34, 42

TransparenciesLesson Transparencies, Volume 4 ....................... Chapter 12

• Teacher Tools• Warm-ups• Teaching Transparencies• Lesson Quizzes

Alternate Openers: Explorations ..........................pp. 84–88

Know-It Notebook ............................................. Chapter 12• Vocabulary• Key Concepts• Graphic Organizers• Chapter Review• Big Ideas

Teacher ToolsPower Presentations

Complete PowerPoint® presentations for Chapter 12 lessons

Lesson Tutorial Videos Holt authors Ed Burger and Freddie Renfro present tutorials to support the Chapter 12 lessons.

Teacher’s One-Stop Planner Easy access to all Chapter 12 resources and assessments, as well as software for lesson planning, test generation, and puzzle creation

IDEA Works! Key Chapter 12 resources and assessments modifi ed to address special learning needs

Solutions Key .................................................... Chapter 12

Interactive Answers and Solutions

TechKeys Lab Resources

Project Teacher Support Parent Resources

Virtual File Cabinet

WorkbooksHomework and Practice Workbook

Teacher’s Edition ...........................................pp. 84–88

Know-It Notebook Teacher’s Guide ........................................... Chapter 12

Review for Mastery Workbook Teacher’s Guide ........................................... Chapter 12

Focus on CAHSEE Standards: Intervention Workbook Teacher’s Guide


858D

Lesson Tutorial Videos Multilingual Glossary Online Interactivities Starring Holt authors Ed Burger and Freddie Renfro! Live tutorials to support every lesson in Chapter 12.

Searchable glossary includes defi nitions in English, Spanish, Vietnamese, Chinese, Hmong, Korean, and other languages.

Interactive tutorials provide visually engaging alternative opportunities to learn concepts and master skills.

Technology Highlights for Universal Access

Universal Access

KEY: SE = Student Edition TE = Teacher’s Edition CRF = Chapter Resource File Spanish available Available online Available on CD-ROM

Strategic StudentsPractice A ................................................CRF, every lesson

Review for Mastery .................................CRF, every lesson

Reading Strategies ...................................CRF, every lesson

Academic Vocabulary Connections .........................TE p. 859

Visual Cues ................................................TE pp. 872, 881

Questioning Strategies ............................ TE, every example

Ready to Go On? Intervention ............................ Chapter 12

Know-It Notebook ............................................ Chapter 12

Homework Help Online

Lesson Tutorial Videos

Online Interactivities

Special Needs StudentsPractice A ................................................CRF, every lesson

Review for Mastery .................................CRF, every lesson

Reading Strategies ...................................CRF, every lesson

Academic Vocabulary Connections .........................TE p. 859

Inclusion ....................................................TE pp. 891, 894

IDEA Works! Modifi ed Resources ....................... Chapter 12

Ready to Go On? Intervention ............................ Chapter 12

Know-It Notebook ............................................ Chapter 12

Lesson Tutorial Videos

Online Interactivities

English LearnersReading Strategies ...................................CRF, every lessonVocabulary Exercises .........................SE, every exercise setAcademic Vocabulary Connections .........................TE p. 859English Language Learners .........TE pp. 861, 871, 894, 906,

922–923Summary and Review ........................... Chapter 12Success for English Language Learners ............... Chapter 12Know-It Notebook ............................................. Chapter 12Multilingual Glossary Lesson Tutorial Videos

Benchmark StudentsPractice B ................................................CRF, every lessonProblem Solving ......................................CRF, every lessonAcademic Vocabulary Connections .........................TE p. 859Questioning Strategies ............................ TE, every exampleReady to Go On? Intervention ............................ Chapter 12Know-It Notebook ............................................. Chapter 12Homework Help Online Online Interactivities

Advanced StudentsPractice C ................................................CRF, every lessonChallenge ................................................CRF, every lessonReading and Writing Math EXTENSION .......................TE p. 861Concept Connection EXTENSION ......................TE pp. 888, 908Advanced Learners/GATE ......................................TE p. 901Ready to Go On? Enrichment ............................. Chapter 12

ENGLISH LANGUAGE LEARNERS

Teaching Tips to help all students appear throughout the chapter. A few that target specifi c students are highlighted below.


858E Chapter 12

Ongoing Assessment

KEY: SE = Student Edition TE = Teacher’s Edition AR = Assessment Resources Spanish available Available online Available on CD-ROM

Are You Ready? Ready to Go On? Focus on CAHSEE Standards: Benchmark Tests and Intervention

Automatically assess readiness and prescribe intervention for Chapter 12 prerequisite skills.

Automatically assess understanding of and prescribe intervention for Sections 12A and 12B.

Automatically assess profi ciency with California Algebra 2 Standards and provide intervention.

Technology Highlights for Ongoing Assessment

Assessing Prior KnowledgeDetermine whether students have the prerequisite concepts and skills for success in Chapter 12.

Are You Ready? .................................... SE p. 859Warm Up ................................... TE, every lesson

Chapter and Standards AssessmentProvide review and practice for Chapter 12 and standards mastery.

Concept Connection ................................... SE pp. 888, 908Study Guide: Review .................................. SE pp. 912–915Standardized Test Prep .............................. SE pp. 920–921

College Entrance Exam Practice ............................ SE p. 917Focus on CAHSEE Standards: Benchmark Tests Focus on CAHSEE Standards: Intervention Workbook

California Standards Practice CD-ROM CAHSEE Standards Practice CD-ROM IDEA Works! Modifi ed Worksheets and Tests

Alternative AssessmentAssess students’ understanding of Chapter 12 concepts and combined problem-solving skills.

Alternative Assessment .............................. TE, every lessonPerformance Assessment ........................... AR pp. 239–240Portfolio Assessment ......................................... AR p. xxxivChapter 12 Project

Daily AssessmentProvide formative assessment for each day of Chapter 12.

Questioning Strategies ............................ TE, every exampleThink and Discuss ......................................SE, every lesson

Check It Out! Exercises ............................SE, every exampleWrite About It ...........................................SE, every lessonJournal ...................................................... TE, every lesson

Lesson Quiz ............................... TE, every lessonAlternative Assessment .............................. TE, every lessonIDEA Works! Modifi ed Lesson Quizzes ............... Chapter 12

Weekly AssessmentProvide formative assessment for each week of Chapter 12.

Concept Connection ................................... SE pp. 888, 908Ready to Go On? ........................ SE pp. 889, 909Cumulative Assessment ............................. SE pp. 920–921Test and Practice Generator .............One-Stop Planner

Formal AssessmentProvide summative assessment of Chapter 12 mastery.

Section Quizzes ......................................... AR pp. 225–226Chapter 12 Test .................................................... SE p. 916Chapter Test (Levels A, B, C) ....................... AR pp. 227–238

• Multiple Choice • Free Response

Cumulative Test ......................................... AR pp. 241–244Test and Practice Generator .............One-Stop Planner

A Chapter 12 Test

C Chapter 12 Test

A Chapter 12 Test

C Chapter 12 Test

B Chapter 12 Test

MULTIPLE CHOICE

B Chapter 12 Test

FREE RESPONSE

12. What is the common difference of the sequence 3, 4.5, 6, 7.5, …?

A 1.5 C 3

B 2

13. What is S6 for the arithmetic series 4 � 4.2 � 4.4 � 4.6 � 4.8 � …?

A 5 B 27

14. A new restaurant has only 20 customers during their first week open. If they increase the number of customers by 15% each week, approximately how many customers will they have during the eighth week?

A 44 C 61

B 53

15. What is the geometric mean of 4 and 16?

A �8 B �10

16. What is the common ratio of the

geometric sequence 3, 3__2

, 3__4

, 3__8

, ….

A 1__3

C 3__2

B 1__2

17. Marsha saves $92 during her first month at a new job. If she increases the amount she saves by 20% each month, how much will she have saved after 3 months?

A $312.80 B $334.88

18. What is the sum of the geometric series

� m�1

3

2 � 3 �m�1 ?

A 15 C 26

B 18

19. A scouting troop eats 64 hot dogs on the first night of their camping trip. They eat only half as many as the night before each night. If the trip last 6 days, how many hot dogs do they eat in all?

A 126 B 224

20. Which of the following series diverges?

A 10 � 5 � 2.5 � 1.25 � …

B 100 � 90 � 81 � 72.9 � …

C 10 � 15 � 22.5 � 33.75 � …

21. If you follow the pattern below, how many boxes will you use to complete the display ?

A 18 C 24

B 20

22. Evaluate � q�1

�

� 1__2 �q

.

A 1 B 2

23. Which is a counterexample that disproves 3n � 3n?

A n � �1 C n � 1

B n � 0

24. An infinite geometric series has a sum of 10 and a common ratio of 0.5. What is the first term of the series?

A 1 C 20

B 5

Select the best answer.

1. What are the first 5 terms of the sequence where a1 � 2 andan � �2an�1 ?

A 2, 4, 8, 16, 32

B 2, �4, 8, �16, 32

C �2, 4, �8, 16, �32

2. A company is tracking the number of complaints received.During the first 4 months, they record the following numbers of complaints: 20, 25, 30, and 35. Which is a possible explicit rule for the number of complaints they will receive in the nth month?

A an � 20n � 5 B an � 15 � 5n

3. Jeremiah puts $950 in a savings account earning 5% compounded annually. If he doesn’t withdraw or make any more deposits, approximately how much is his savings account worth after 3 years?

A $995 C $1100

B $1092

4. How many black triangles are in the next two iterations of the sequence shown below?

A 6, 9 C 10, 15

B 9, 12

5. Which is summation notation for the series 2 � 4 � 8 � 16?

A � k�1

4

2k B � k�1

4

2k

6. The first day, Jervaise’s paper route took 100 minutes to complete. Each day takes 10% less than the day before. How many minutes, total, does Jervaise work during the first 3 days?

A 90 C 271

B 270

7. Evaluate � k�1

11

4.

A 4 C 44

B 11

8. Brittany has a 25 page report to write. If every day she writes 20% of the pages that remain unwritten, how many pages will she write in 2 days?

A 5 C 9

B 8

9. Evaluate � k�1

4

� 2 �k�1 .

A 8 C 16

B 15

10. Kieran practices piano 30 minutes the first day and adds 10 minutes each day thereafter. How many minutes does Kieran practice, total, over 4 days?

A 150 B 180

11. Valentine increases the number of books she reads by the same amount each month. If she reads 4 books in the second month and 8 books in the sixth month, how many books does she read in the third month?

A 5 C 7

B 6

Chapter 12 Test

MODIFIED FOR IDEA


858F

Formal Assessment

Three levels (A, B, C) of multiple-choice and free-response chapter tests are available in the Assessment Resources.

Create and customize Chapter 12 Tests. Instantly

generate multiple test versions, answer keys, and

practice versions of test items.

Chapter 12 Test(continued)

B Chapter 12 Test (continued)

B Chapter 12 Test (continued)

Modified tests and worksheets found in IDEA Works!

858 Chapter 12

12A Exploring Arithmetic Sequences and Series

12-1 Introduction to Sequences

12-2 Series and Summation Notation

Lab Evaluate Sequences and Series

12-3 Arithmetic Sequences and Series

12B Exploring Geometric Sequences and Series

12-4 Geometric Sequences and Series

Lab Explore Infinite Geometric Series

12-5 Mathematical Induction and Infinite Geometric Series

Ext Area Under a Curve

KEYWORD: MB7 ChProj

Sunflowers grown in California are used as floral decorations

and to make birdseed and vegetable oil.

Sequences and Series

858 Chapter 12

S E C T I O N 12AExploring Arithmetic Sequences and Series

On page 888, stu-dents use arithmetic sequences and series to analyze

tetrahedral kites.

Exercises designed to prepare students for success on the Concept Connection can be found on pages 867, 876, and 886.

S E C T I O N 12BExploring Geometric Sequences and Series

On page 908, stu-dents use geometric sequences and series to analyze and predict

the box office revenue of hit movies.

Exercises designed to prepare students for success on the Concept Connection can be found on pages 897 and 906.

Golden RectanglesIn the Chapter Project, students explore the relationship between the Fibonacci sequence and the golden ratio. Then they construct a golden rectangle and discover how its proportions relate to the golden ratio.

About the Project

Project ResourcesAll project resources for teachers and students are provided online.

Materials:• calculator• compass and straightedge (MK)

KEYWORD: MB7 ProjectTS

Exploring patterns, sequences, and series is an important part of

mathematics. Sequences are vis-ible all around us. For example, the Fibonacci sequence can be seen in the spirals created by the seeds in a sunflower. Students can explore the Fibonacci sequence in Lesson 12-1 and apply it to geometric forms in the Chapter Project.

Algebra in California

Sequences and Series 859

VocabularyMatch each term on the left with a definition on the right.

1. exponential function

2. function

3. linear equation

4. quadratic function

A. a pairing in which there is exactly one output value for each input value

B. an equation whose graph is a straight line

C. a function defined by a quotient of two polynomials

D. a function of the form f (x) = a x 2 + bx + c, where a ≠ 0

E. a function of the form f (x) = a b x, where a ≠ 0 and b ≠ 1

Simplify Radical ExpressionsSimplify each expression.

5. √ � 25 · √ � 36 6. √ �� 121 -

√ � 81 7. √

��

1_49

8. √ � 16_√ � 64

Evaluate PowersEvaluate.

9. (-3) 3 10. (-5) 4 11. 1 - (- 23)

312. 2 2 · 2 7_

( 2 2) 5

Solve for a VariableSolve each equation for x.

13. y = 12x - 5 14. y = -

x_3

+ 1

15. y = -9 + x2 16. y = -4(x2- 9)

Evaluate ExpressionsEvaluate each expression for x = 2, y = 12, and z = 24.

17. y (y + 1)_

3x 18. z + (y - 1)x

19. y(

x + z_2 )

20. z (

1 - y_1 - x)

CounterexamplesFind a counterexample to show that each statement is false.

21. n2= n, where n is a real number 22. n3

≥ n2≥ n, where n is a real number

23. 1_n >

1_ n 2

, where n is a real number 24. 2_n ≠

n_2

, where n is a real number

E

B

A

D

30 21_7

1_2

-27 625 513

1_2

x = y + 5_

12x = -3y + 3

x = ± √ �� 36 - y_

2x = ± √ �� y + 9

26 46

264156

2 2 ≠ 2( 1_

2 )3

< ( 1_2 )

2< ( 1_

2 )1

1_1 = 1_

1 2 or 1_

-2 < 1_

(-2) 22_2 = 2_

2

Possible answers given.

Are You Ready? 859

NOINTERVENE

YESENRICHDiagnose and Prescribe

ARE YOU READY? Intervention, Chapter 12

Prerequisite Skill Worksheets CD-ROM Online

Simplify Radical Expressions Skill 53 Activity 53

Diagnose and Prescribe Online

Evaluate Powers Skill 8 Activity 8

Solve for a Variable Skill 72 Activity 72

Evaluate Expressions Skill 60 Activity 60

Counterexamples Skill 89 Activity 89

OrganizerObjective: Assess students’ understanding of prerequisite skills.

Prerequisite Skills

Simplify Radical Expressions

Evaluate Powers

Solve for a Variable

Evaluate Expressions

Counterexamples

Assessing Prior Knowledge

INTERVENTIONDiagnose and Prescribe

Use this page to determine whether intervention is necessary or whether enrichment is appropriate.

ResourcesAre You Ready? Intervention and Enrichment Worksheets

Are You Ready? CD-ROM

Are You Ready? Online

ARE YOU READY? Enrichment, Chapter 12

WorksheetsCD-ROMOnline

860 Chapter 12

California Standard

Academic Vocabulary Chapter Concept

21.0 Students apply the method of mathematical induction to prove general statements about the positive integers.

(Lesson 12-5)

method a way of doing something

general applying to every member of a group

You prove that formulas work for all natural numbers.

22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

(Lessons 12-3, 12-4, 12-5)

arithmetic series the sum of the terms of a sequence in which successive terms differ by the same number

Example:

5 + 8 + 11 + 14 + 17

geometric series the sum of the terms of a sequence in which the ratio of successive terms is constant

Example:

1 + 1_2 + 1_4 + 1_8 + 1_16

You fi nd an expression that describes any term in a sequence.

23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

(Lessons 12-3, 12-4, 12-5)(Lab 12-5)

derive to reach a conclusion by reasoning

summation the act or process of adding

You develop and use formulas to fi nd the sum of a series.

The information below “unpacks” the standards. The Academic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter.

860 Chapter 12

OrganizerObjective: Help students organize the new concepts they will learn in Chapter 12.

Online EditionMultilingual Glossary

Academic Vocabulary ConnectionsTo become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.

1. What does the word sequence mean in everyday usage? What might a number sequence refer to? Sequence relates to the order in which things are arranged. A number sequence might be numbers placed in order.

2. The word finite means “having definite or definable limits.” Give examples of sentences that use the word finite. Explain what a finite sequence might refer to. There are a finite number of possible sums on a pair of number cubes. A finite sequence has an end.

3. Using the previous definition of finite, give examples of sentences that use the word infinite. Explain what an infinite sequence might refer to. You can combine colors in an infinite number of ways to create a painting. An infinite sequence continues without end.

C H A P T E R

12Unpacking the Standards

Previously, students

• studied sets of numbers, including natural numbers and perfect squares.

• used patterns of differences or ratios to classify data.

• graphed and evaluated linear and exponential functions.

Students will study

• patterns of numbers, called sequences, and their sums, called series.

• patterns to determine whether sequences are arithmetic or geometric.

• how to write and evaluate sequences and series.

Students can use these skills

• in future math classes, especially Precalculus and Calculus.

• in Physics classes to model patterns, such as the heights of bouncing objects.

• to calculate the growth of financial investments.

Sequences and Series 861

Writing Strategy: Write a Convincing ArgumentBeing able to write a convincing argument about a math concept shows that you understand the material well. You can use a four-step method to write an effective argument as shown in the response to the exercise below.

Try This

Use the four-step method described above to answer each question.

1. A number cube is rolled 20 times and lands on the number 3 twice. What is the fewest number of rolls needed for the experimental probability of rolling a 3 to equal the theoretical probability of rolling a 3? Explain how you got your answer.

2. Aidan has narrowed his college choices down to 9 schools. He plans to visit 3 or 4 schools before the end of his junior year. How many more ways can he visit a group of 4 schools than a group of 3 schools? Explain.

Identify the goal.

The goal is to describe the difference between theoretical and experimental probability.

Provide a statement of response to the goal.

Theoretical probability is based purely on mathematics, but experimental probability is based on the results of an experiment.

Provide the evidence to support your statement.

In a coin toss, the theoretical probability of tossing heads

is number of favorable outcomes_____________________________number of outcomes in the sample space

= 1__2

.

The experimental probability of tossing heads is number of times the event occurs_______________________number of trials

.

In one trial, the result is either heads or tails, so the experimental probability will be 1 or 0. The theoretical probability is still 1__

2 .

Summarize your argument.

Because theoretical probability is based solely on the mathematical outcomes, it never changes. Experimental probability is based on actual results, so it may change with each trial of an experiment.

Step 1

Step 2

Step 3

Step 4

35. Write About It Describe the difference between theoretical probability and experimental probability. Give an example in which they may differ.

From Lesson 11-2

Reading and Writing Math 861

OrganizerObjective: Help students apply strategies to understand and retain key concepts.

Online Edition

ResourcesChapter 12 Resource File

Reading Strategies

Writing Strategy:


Write a Convincing ArgumentDiscuss Encourage students to suggest reasons why they might benefit from writing a convincing argument about a math concept. Help them realize that clearly articulating math concepts will enable them to understand the concepts more fully.

Extend As students work through Chapter 12, have them explain the new concepts in their own words. Suggest that the class work together to provide explanations that are as complete as possible. Ask students to suggest connections to concepts they have already learned and tell how those concepts are extended by the new material

Answers 1. Step 1: Find the least number

of rolls for the experimental and theoretical probabilities to be equal. Step 2: Both probabilities are ratios that can be compared.

Step 3: Theoretical P (3) = 1 _ 6 ,

which is equivalent to 2 _ 12

,

3 _ 18

, 4 _ 24

, etc. After 20 rolls, the

first possible opportunity for the probabilities to be equal is

at 4 _ 24

, so at least 4 more rolls

are needed.

Step 4: If the number 3 is rolled 2 out of the next 4 rolls, the experimental probability would

be 4 _ 24

, which is equal to the

theoretical probability, 1 _ 6 .

2. See p. A46.

C H A P T E R

12

One-Minute Section PlannerLesson Lab Resources Materials

Lesson 12-1 Introduction to Sequences • Find the nth term of a sequence.

• Write rules for sequences.

□✔ SAT-10 □✔ NAEP

□✔ ACT □✔ SAT

□✔ SAT Subject Tests

Algebra Lab Activities12-1 Algebra Lab

Optionalgraphing calculator

Lesson 12-2 Series and Summation Notation • Evaluate the sum of a series expressed in sigma notation.

□✔ SAT-10 □

NAEP □

ACT □✔ SAT



12-2 Technology Lab Evaluate Sequences and Series • Use a graphing calculator to generate the terms of a sequence and

find the sums of series.




Technology Lab Activities12-2 Lab Recording Sheet

Requiredgraphing calculator

Lesson 12-3 Arithmetic Sequences and Series • Find the indicated terms of an arithmetic sequence.

• Find the sums of arithmetic series.




Optionalcentimeter squares or cubes (MK), graphing calculator

MK = Manipulatives Kit

SECTION

12A

862A Chapter 12

Exploring Arithmetic Sequences and Series

862B

SEQUENCESLesson 12-1A sequence can be defined as a function a whose domain is the natural numbers (or a subset thereof) and whose range is another set of numbers. We often represent a(n) by an and write the sequence as an ordered list:

• a1, a2, a3, … (for infinite sequences)• a1, a2, a3, …, an (for finite sequences)

Consider the sequence an =

1__2n2, which begins 1__

2 , 2,

41__2, 8,... . As a function, the sequence maps 1 to 1__

2, 2

to 2, 3 to 41__2, and 4 to 8. These points are plotted on

the coordinate plane below. The graph shows that the

points lie on the curve y =

1__2x2. In other words, the

sequence an =

1__2n2 is precisely the function y =

1__2x2

with its domain restricted to the natural numbers.

However, it is important to understand that many sequences are not given by simple algebraic formulas and in fact may appear to be just random sequences of digits.

SERIESLesson 12-2A series is the sum of the terms of a sequence. A series may add a finite number of terms or an infinite num-ber of terms. A partial sum, Sn, is the sum of the first nterms of a sequence.

A well-known story about the mathematician Carl Friedrich Gauss (1777–1885) states that a harried teacher once asked Gauss and his classmates to add all of the natural numbers from 1 to 100. To the teacher’s surprise, this task did not keep all the students quiet

for very long; Gauss immediately gave the correct answer, 5050. He saw that the terms of the series can be rearranged to form 50 pairs that add to 101:

1 + 100 = 101 2 + 99 = 101 3 + 98 = 101

� �

50 + 51 = 101

This is the key idea behind the formula for the sum of a linear series:

∑

k=1

n

k =

n(n + 1)_2

ARITHMETIC SEQUENCES AND SERIES 22.0, 23.0

Lesson 12-3In an arithmetic sequence, successive terms differ by the same nonzero number d, called the constant differ-ence. Because the nth term can be found by adding d (n - 1) times to the first term, the general rule for the nth term of an arithmetic sequence is

an = a1 + (n - 1)d.

An arithmetic sequence can be viewed as a linear function whose domain is restricted to the natural numbers. In fact, the general rule above has much in common with the equation of a line in slope-intercept form, y = mx + b. In particular, the constant difference d is the slope of the line through the points (n, an).

The sum of the first n terms of an arithmetic series Sn

is given by the formula

Sn = n(

a1 + an_2 ) .

This formula may be developed as follows. The sum of the series Sn divided by the number of terms n is the

average of the terms. The average of the terms must

be a1 + an______2 since the terms are equally spaced and sym-

metric about this value on a number line. Thus, Sn__n =

a1 + an______2 , which immediately yields the required

formula.

Math Background: Teaching the Standards

Who uses this?Sequences can be used to model many natural phenomena, such as the changes in a rabbit population over time.

862 Chapter 12 Sequences and Series

In 1202, Italian mathematician Leonardo Fibonacci described how fast rabbits breed under ideal circumstances. Fibonacci noted the number of pairs of rabbits each month and formed a famous pattern called the Fibonacci sequence.

A sequence is an ordered set of numbers. Each number in the sequence is a term of the sequence . A sequence may be an infinite sequence that continues without end, such as the natural numbers, or a finite sequence that has a limited number of terms, such as

⎧

⎨

⎩

1, 2, 3, 4 ⎫

⎬

⎭

.

You can think of a sequence as a function with sequential natural numbers as the domain and the terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1 , the second term is a2 , and the nth term is an. Because a sequence is a function, each number n has only one term value associated with it, an.

n 1 2 3 4 5

an 1 1 2 3 5

Term number

Term value

Domain

Range

In the Fibonacci sequence, the first two terms are 1 and each term after that is the sum of the two terms before it. This can be expressed by using the rule a1 = 1, a2 = 1, and an = an-2 + an-1 , where n ≥ 3. This is a recursive formula.A recursive formula is a rule in which one or more previous terms are used to generate the next term.

1E X A M P L E Finding Terms of a Sequence by Using a Recursive Formula

Find the first 5 terms of the sequence with a1 = 5 and an = 2 an-1 + 1 for n ≥ 2.

The first term is given, a 1 = 5.

Substitute a1 into the rule to find a2.

Continue using each term to find the next term.

The first 5 terms are 5, 11, 23, 47, and 95.

n 2an-1 + 1 an

12345

Given2(5) + 12(11) + 12(23) + 12(47) + 1

5 11 23 47 95

Find the first 5 terms of each sequence.

1a. a1 = -5, an = an-1 - 8 1b. a1 = 2, an = -3an-1

12-1 Introduction to Sequences

ObjectivesFind the nth term of a sequence.

Write rules for sequences.

Vocabularysequenceterm of a sequenceinfinite sequencefinite sequencerecursive formulaexplicit formulaiteration

an is read “a sub n.”

California Standards

Preparation for 22.0 Studentsfind the general term and the sums of arithmetic series and of both finite and infinite geometric series.

-5, -13, -21, -29, -37 2, -6, 18, -54, 162

862 Chapter 12

Introduce1

12-1 OrganizerObjectives: Find the nth term of a sequence.

Write rules for sequences.

Algebra LabIn Algebra Lab Activities

Online EditionTutorial Videos

Warm UpEvaluate.

1. (-1) 8 1 2. (11) 2 121

3. (-9) 3 -729 4. (3) 4 81

Evaluate each expression for x = 4.

5. 2x + 1 9

6. 0.5x + 1.5 3.5

7. x 2 - 1 15

8. 2 x + 3 19

Also available on transparency

Q: Why did the fractal lose the dance contest?

A: Its third step was out of sequence.

MotivateAsk students to identify the next 3 terms in the pattern 2, 4, 6, 8, . . . . 10, 12, 14 Discuss how they decided on their answers. Then ask students to identify the next 3 terms in the pattern 2, 4, 8, 16, . . . . 32, 64, 128 Discuss how they decided on their answers. Explain that number patterns like these that follow a rule are called sequences.

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.California Standards

Preparation for 22.0

12-1 Introduction to Sequences 863

In some sequences, you can find the value of a term when you do not know its preceding term. An explicit formula defines the nth term of a sequence as a function of n.

2E X A M P L E Finding Terms of a Sequence by Using an Explicit Formula

Find the first 5 terms of the sequence an = 2 n - 3.

Make a table. Evaluate the sequence for n = 1 through n = 5.

n 2n - 3 an

12345

21 - 322 - 323 - 324 - 325 - 3

-115

1329

The first 5 terms are -1, 1, 5, 13, and 29.

Check Use a graphing calculator. Enter y = 2 x - 3 and make a table.


2a. an = n2- 2n 2b. an = 3n - 5

You can use your knowledge of functions to write rules for sequences.

3E X A M P L E Writing Rules for Sequences

Write a possible explicit rule for the nth term of each sequence.

A 3, 6, 12, 24, 48, …

Examine the differences and ratios.Ratios 2 2 2 2

Terms 3 6 12 24 48

1st differences 3 6 12 242nd differences 3 6 12The ratio is constant. The sequence is exponential with a base of 2.Look for a pattern with powers of 2. a 1 = 3 = 3 (2)

0 , a 2 = 6 = 3 (2)

1 , a 3 = 12 = 3 (2)

2 , . . .A pattern is 3 (2)

n-1 . One explicit rule is an = 3 (2)

n-1 .

B 2.5, 4, 5.5, 7, 8.5, …

Examine the differences.

Terms 2.5 4 5.5 7 8.5

1st differences 1.5 1.5 1.5 1.5 The first differences are constant, so the sequence is linear.The first term is 2.5, and each term is 1.5 more than the previous.

A pattern is 2.5 + 1.5 (n - 1) , or 1.5n + 1. One explicit rule isan = 1.5n + 1.


3a. 7, 5, 3, 1, -1, … 3b. 1, 1_2

, 1_3

, 1_4

, 1_5

, …

Linear patterns have constant first differences. Quadratic patterns have constant second differences. Exponential patterns have constant ratios.(Lesson 9-6)

-2, 1, 4, 7, 10-1, 0, 3, 8, 15

an = 9 - 2nan = 1_n

Lesson 12-1 863

Example 1

Find the first 5 terms of the sequence with a 1 = -2 and a n = 3 a n - 1 + 2 for n ≥ 2.

-2, -4, -10, -28, -82

Example 2

Find the first 5 terms of the sequence a n = 3 n - 1.

2, 8, 26, 80, 242

Example 3


A. 5, 10, 20, 40, 80, … a n = 5 ( 2 n - 1 )

B. 1.5, 4, 6.5, 9, 11.5, … a n = 2.5n - 1


Additional Examples

INTERVENTION Questioning Strategies

EXAMPLE 1

• Is the sequence increasing or decreasing?

• What effect would a negative multiplier have on the sequence? Explain.

EXAMPLE 2

• How is using an explicit formula different from using a recursive formula?

EXAMPLE 3

• How do differences or ratios help identify the explicit rule for the nth term?

Math Background Remind students of functions they

have already studied this year (linear, quadratic, and exponential)and of regression techniques to find the rules for sequences. You may wish to review Lesson 9-6.Guided Instruction

Compare sequences with the functions students have already learned. Choose a simple linear function, such as f (x) = 5x. Have students evaluate the function for several consecutive integer values. Ask them to describe the results. counting by 5’s or multiples of 5 Use the answer to explain the difference between recursive and explicit formulas.

Teach2

Through Cognitive Strategies

Students may need help with the new notation. Stress that n is the term number (also called the index of a term) and a n is the nth term, so a n - 1 is the previous term and a n + 1 is the next term. To reinforce these concepts, have students identify some terms for a given n-value. For example, which term is a n + 2 when n = 3? fifth


4E X A M P L E Physics Application

A ball is dropped and bounces to a height of 5 feet. The ball rebounds to 60% of its previous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 9th bounce?

Because the ball first bounces to a height of 5 feet and then bounces to 60% of its previous height on each bounce, the recursive rule is a1 = 5 and an = 0.6 an-1 . Use this rule to find some other terms of the sequence and graph them.

a2 = 0.6 (5) = 3

a3 = 0.6(3) = 1.8

a4 = 0.6(1.8) = 1.08

The graph appears to be exponential. Use the pattern to write an explicit rule.

an = 5 (0.6)

n-1 , where n is the bounce number

Use this rule to find the bounce height for the 9th bounce.

a9 = 5 (0.6)

9-1≈ 0.084 foot, or approximately 1 inch.

The ball is about 0.084 feet high on the 9th bounce.

4. An ultra-low-flush toilet uses 1.6 gallons every time it is flushed. Graph the sequence of total water used after n flushes, and describe its pattern. How many gallons have been used after 10 flushes?

Recall that a fractal is an image made by repeating a pattern (Lesson 5-5). Each step in this repeated process is an iteration , the repetitive application of the same rule.

5E X A M P L E Iteration of Fractals

The Sierpinski triangle is a fractal made by taking an equilateral triangle, removing an equilateral triangle from the center, and repeating for each new triangle. Find the number of triangles in the next 2 iterations.

By removing the center of each triangle, each iteration turns every triangle into 3 smaller triangles. So the number of triangles triples with each iteration.

The number of triangles can be represented by the sequence an = 3 n-1 .

The 4th and 5th terms are a4 = 3 4-1= 27 and a5 = 3 5-1

= 81.

The next two iterations result in 27 and 81 triangles.

5. The Cantor set is a fractal formed by repeatedly removing the middle third of a line segment as shown. Find the number of segments in the next 2 iterations.

Do not connect the points on the graph because the number of bounces is limited to the set of natural numbers.

The graph shows the points lie on a line with positive slope; 16 gal.

8, 16

864 Chapter 12

Example 4

A ball is dropped and bounces to a height of 4 feet. The ball rebounds to 70% of its previous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 10th bounce?

a n = 4 (0.7) n - 1 ; ≈ 0.161 ft, or ≈ 2 in.

Example 5

Find the number of triangles in the 7th and 8th iterations of the Sierpinski triangle. 729, 2187


Additional Examples


EXAMPLE 4

• How does the graph help identify the rule?

• Why would you find an explicit rule if you know the recursive rule?

EXAMPLE 5

• Why might you use a sequence to represent the iterations when you can draw pictures?

Number Sense To help students understand the discrete nature of the situ-

ation in Example 4, point out that 1.5 bounces would be impossible.

Math Background Example 4 is an exam-ple of Zeno’s paradox.

According to the formula, the ball will never stop bouncing. Have students research this topic as an extension or challenge activity.

Assess After the Lesson12-1 Lesson Quiz, TE p. 868

Alternative Assessment, TE p. 868

Monitor During the LessonCheck It Out! Exercises, SE pp. 862–864Questioning Strategies, TE pp. 863–864

Diagnose Before the Lesson12-1 Warm Up, TE p. 862

and INTERVENTIONSummarizeAsk students to define a sequence. A sequence is an ordered set of numbers. Then have them explain how to use a recursive formula and an explicit formula to find the terms of a sequence. A recur-sive formula uses previous terms to gener-ate successive terms. An explicit formula defines the nth term of a sequence as a function of n.

Close3


GUIDED PRACTICE 1. Vocabulary A formula that uses one or more previous terms to find the next term

is a (n)

−−−

? formula. (explicit or recursive)

SEE EXAMPLE 1 p. 862


2. a1 = 1, an = 4 an-1 - 1 3. a1 = 3, an = an-1 + 11 4. a1 = 500, an = an-1_

5


5. an = 12 (n - 2) 6. an = (-

1_2)

n-17. an = -3n2

8. an = n(n - 1) 9. an = 4 n-1 10. an = (n + 1)

2



11. 6, 9, 12, 15, 18, … 12. 1_2

, 2_3

, 3_4

, 4_5

, 5_6

, … 13. 25, 15, 5, -5, -15, …


14. Income Billy earns $25,000 the first year and gets a 5% raise each year. Graph the sequence, and describe its pattern. How much will he earn per year after 5 years? 10 years?


15. Patterns Find the number of segments in the next 2 terms of the pattern shown.

PRACTICE AND PROBLEM SOLVINGFind the first 5 terms of each sequence.

16. a1 = 7, an = an-1 - 3 17. an = 1_n2

18. a1 = 4, an = 1.5 an-1 - 2

19. an = (2)

n-1+ 8 20. an = 2 n2

- 12 21. a1= -2, an = -3an-1 - 1

THINK AND DISCUSS 1. Explain the difference between a recursive rule and an explicit rule.

2. Identify three possible next terms for the sequence 1, 2, 4, . . . .

3. Describe how a sequence is a function. Do all sequences have the same domain? Explain.

4. GET ORGANIZED Copy and complete the graphic organizer. Summarize what you have learned about sequences.

12-1 ExercisesExercisesKEYWORD: MB7 12-1

KEYWORD: MB7 Parent

California StandardsPrep for 22.0; 7.0

recursive

1, 3, 11, 43, 171

3, 14, 25, 36, 47500, 100, 20, 4, 4_

5

0, 2, 6, 12, 20 1, 4, 16, 64, 256 4, 9, 16, 25, 36

an = 3 + 3n an = 35 - 10n

exponential growth; about $31,907; about $40,722

16, 32

7, 4, 1, -2, -5 4, 4, 4, 4, 4

-2, 5, -16, 47, -1429, 10, 12, 16, 24 -10, -4, 6, 20, 38

1, 1_4 , 1_

9 , 1_

16 , 1_

25

12. an = n_n + 1

Lesson 12-1 865

Answers to Think and DiscussPossible answers:

1. A recursive rule uses one or more previous terms to find the value of a term. An explicit rule uses the first term and the term number to find the value.

2. 1, 7, 8

3. There is only one output, a n , associated with each input, n. The domain of a sequence is the set of term numbers, but sequences do not necessarily have the same domain.

4. See p. A13.

Assignment Guide

Assign Guided Practice exercises as necessary.

If you finished Examples 1–2 Basic 16–21, 27–29 Proficient 16–21, 27–31 Advanced 16–21, 27–32

If you finished Examples 1–5 Basic 16–29, 33–36, 43–46,

49–54, 59–66 Proficient 16–46, 49–54, 59–66 Advanced 16–33, 34–46 even,

47–66

Homework Quick CheckQuickly check key concepts.Exercises: 16, 20, 22, 25, 26, 28

12-1 ExercisesExercises

Technology Students are taught to use the seq( function on graphing calculators in the

Technology Lab following Lesson 12-2. You may want to teach the first part of that lab before or during this lesson.

Answers 5. -12, 0, 12, 24, 36

6. 1, - 1 _ 2 , 1 _

4 , - 1 _

8 , 1 _

16

7. -3, -12, -27, -48, -75

14.

Single-Play Chess Tournament

Number of Players Number of Games

1 0

2 1

3 3

4 6

5 10


The Fibonacci sequence can also be used to discover the golden ratio. The ratios of successive terms become closer and closer to the golden ratio, 1 + √ � 5_____

2 .

Math History

For See Exercises Example

16–18 1 19–21 2 22–24 3 25 4 26 5

Independent Practice Write a possible explicit rule for the nth term of each sequence.

22. 2, 8, 18, 32, 50, . . . 23. 9, 5, 1, -3, -7, … 24. 5, 0.5, 0.05, 0.005, …

25. Architecture Chairs for an orchestra are positioned in a curved form with the conductor at the center. The front row has 16 chairs, and each successive row has 4 more chairs. Graph the sequence and describe its pattern. How many chairs are in the 6th row?

26. Fractals Find the number of squares in the next 2 iterations of Cantor dust as shown.


27. a1 = 12, an =

1_2

an-1 + 2 28. a1 = 1, an =

2_an-1

29. a1 = -10, an = - an-1 + 10

30. an = 2 n2- 12 31. an = 8 -

1_10

n 32. an = 5 (-1)

n+1(3)

n-1

33. /////ERROR ANALYSIS///// Two attempts to find the first 5 terms of the sequence a1 = 3 and an = 2n + 1 are shown. Which is incorrect? Explain the error.

Write a possible explicit rule for each sequence, and find the 10th term.

34. 16, 4, 1, 1_4

, 1_16

, … 35. 15_9

, 14_9

, 13_9

, 12_9

, 11_9

, … 36. -5.0, -2.5, 0, 2.5, 5.0, …

37. 1, -

1_2

, 1_3

, -

1_4

, 1_5

, … 38. 0.04, 0.4, 4, 40, 400, … 39. 24, 21, 16, 9, 0, …

40. Fibonacci Recall from the lesson that the Fibonacci sequence models the number of pairs of rabbits after a certain number of months. The sequence begins 1, 1, . . ., and each term after that is the sum of the two terms before it.

a. Find the first 12 terms of the Fibonacci sequence.

b. How many pairs of rabbits are produced under ideal circumstances at the end of one year?

Find the number of dots in the next 2 figures for each dot pattern.

41. 42.

43. Chess Ronnie is scheduling a chess tournament in which each player plays every other player once. He created a table and found that each new player added more than one game.

a. Graph the sequence and describe its pattern. What are the next 2 terms in the sequence?

b. Use a regression to find an explicit rule for the sequence.

c. What if...? How would the schedule change if each player played every other player twice? Make a table, and describe how the sequence is transformed.

Skills Practice p. S26Application Practice p. S43

Extra Practice

an = 2n2 an = 13 - 4n

linear with a slope of 4; 36

256, 1024

-10, 20, -10, 20, -101, 2, 1, 2, 1

27. 12, 8, 6, 5, 4 1_2

B is incorrect. The formula is explicit, not recursive.

an = 25 - n2 ; -75

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

144 rabbit pairs

15, 21 25, 36

an = 1_2

n2 - 1_2

n

24. an = 5(0.1)n-1

15, 21

866 Chapter 12

Geometry Link In Exercise 26, students may want to draw the

next two iterations of Cantor dust. Point out the difficulty of doing this by hand-there are too many small squares. If students want to visual-ize the iterations, suggest that they instead consider the number of squares formed along the side of the bigger square with each iteration.

Math Background The sequence pictured in Exercise 41 is the

sequence of triangular numbers, and the sequence pictured in Exercise 42 is the sequence of square numbers. Discuss why that is so.

Answers

25.

30. -10, -4, 6, 20, 38

31. 7.9, 7.8, 7.7, 7.6, 7.5

32. 5, -15, 45, -135, 405

34. a n = 16 ( 1 _ 4 )

n-1

; 1 _ 16,384

35. a n = 16 _ 9 - 1 _

9 n; 2 _

3

36. a n = -7.5 + 2.5n; 17.5

37. a n = (-1) n+1

_ n ; - 1 _ 10

38. a n = 0.04 ( 10 n-1 ) ; 40,000,000

12-1 REVIEW FOR MASTERY12-1 READING STRATEGIES

12-1 PRACTICE C

12-1 PRACTICE B

12-1 PRACTICE A


45. Geometry The sum of the interior angle measures for the first 5 regular polygons is shown.

a. Write an explicit rule for the nth term of the sequence of the sum of the angle measures. What is the sum of the measures of the interior angles of a 12-sided regular polygon?

b. Recall that all angles are congruent in a regular polygon. Make a table for the measure of an interior angle for each regular polygon. Graph the data, and describe the pattern.

c. Write an explicit rule for the nth term of the sequence described in part b.

d. Find the measure of an interior angle of a 10-sided regular polygon.

46. Estimation Estimate the 20th term of the sequence 7.94, 8.935, 9.93, 10.925, 11.92, … Explain how you reached your estimate.

47. Music Music involves arranging different pitches through time. The musical notation below indicates the duration of various notes (and rests).

Symbols for Musical Notes and Rests

Whole Half Quarter Eighth Sixteenth Thirty–second

a. Write a numerical sequence that shows the progression of notes (and rests). Write a recursive formula and an explicit formula to generate this sequence.

b. In 4/4 time, a whole note represents 4 beats, a half note represents 2 beats, a quarter note represents 1 beat, and so on. Write a sequence for the number of beats that each note in the progression represents. Then write a recursive formula and an explicit formula to generate this sequence. How is this sequence related to the sequence in part a?

Sum of Interior Angle Measures

44. This problem will prepare you for the Concept Connection on page 888.

A kite is made with 1 tetrahedron in the first (top) layer, 3 tetrahedrons in the second layer, 6 tetrahedrons in the third layer, and so on. Each tetrahedron is made by joining six sticks of equal length.

a. The rule an = an-1 + 6n gives the number of sticks needed to make the nth layer of the kite, where a1 = 6. Find the first five terms of the sequence.

b. Use regression to find an explicit rule for the sequence.

c. How many sticks are needed to build the 10th layer of the kite?

an = 180 (n - 2) for n ≥ 3; 1800°46. Possible answer: about 27; the first term is almost 8. Each term is about 1 more than the previous term and 8 + 19 is 27.

47a. 1, 1_2

, 1_4

, 1_8 , 1_

16 , 1_

32 ;

a 1 = 1, an = 1_2 an -1 ;

an = ( 1_2 )

n-1

47b. 4, 2, 1, 1_2

, 1_4 , 1_

8 ;

a 1 = 4, an = 1_2 an-1 ;

an = 4 ( 1_2 )

n-1 ;

possible answer: it follows the same pattern but starts at a different value.

330

6, 18, 36, 60, 90 a n = 3 n2 + 3n

Lesson 12-1 867

12-1 PROBLEM SOLVING 12-1 CHALLENGE

In Exercises 37 and 54, students might have difficulty working with alternating signs. Remind them that the value of (-1) n is negative when n is odd and positive when n is even.

Exercise 44 involves finding terms of a sequence and writ-

ing a rule for the sequence. This exercise prepares students for the Concept Connection on page 888.

Critical Thinking In Exercise 45, some stu-dents may remember the

formula for the sum of the interior angle measures from geometry. Encourage those students to use the method presented in the problem to verify the formula they remember.

Answers43a. It appears to be quadratic.

c. Players 1 2 3 4 5Games 0 2 6 12 20

The sequence is twice the previ-ous sequence. The function is a vertical stretch by a factor of 2.

45b. Sides Angle Measure (◦) 3 604 905 1086 1207 128 5 _ 7 8 135

It appears to be a square-root or logarithmic curve.

c. a n =180 (n - 2) __

n for n ≥ 3

d. 144°


48. Critical Thinking Can the recursive rule and the explicit rule for a formula ever be the same?

49. Write About It Explain how an infinite sequence is different from a finite sequence.

50. Which is the next term in the sequence -9, -6, -3 , 0, …?

-3 0 3 6

51. Which rule describes the given sequence 4, 12, 36, 108, . . .?

an = 4 + 3n a1 = 4, an = 3 an-1, n ≥ 2

an = 3 + 4n a1 = 3, an = 4 an-1, n ≥ 2

52. Which sequence is expressed by the rule an =

2n_n + 1

?

2_3

, 4_5

, 6_7

, 8_9

, 10_11

, … 0, 1, 2, 3_2

, 8_5

, …

1, 4_3

, 3_2

, 8_5

, 5_3

, … 2, 3_2

, 8_5

, 5_3

, 12_7

, …

53. Which sequence is expressed by the rule a1 = 6 and an = 12 - 2an-1, n ≥ 2?

6, 4, 2, 0, -2, -4, … 6, 0, 12, -12, 36, …

0, 12, -12, 36, -60, … 6, 0, -6, -12, -18, …

54. Gridded Response Find the next term in the sequence -32, 16, -8, 4, -2, . . . .

CHALLENGE AND EXTENDWrite an explicit rule for each sequence and find the 10th term.

55. -

2_3

, 5_3

, 8, 61_3

, 122_3

, … 56. -2, 6, -12, 20, -30, … 57. 0.9, 0.8, 0.6, 0.3, -0.1, …

58. Geometry Draw 5 circles. Place 1 point on the first circle, 2 points on the second, 3 points on the third, and so forth. Then connect every point with every other point in each circle, and count the maximum number of nonoverlapping regions that are formed inside each circle.

a. Write the resulting sequence.

b. Although the sequence appears to double, the sixth circle has less than 32 possible regions. Try to find them all by carefully drawing this figure. How many did you get?

SPIRAL REVIEWSimplify. Assume that all expressions are defined. (Lesson 8-2)

59. x2- 9__

x2+ 5x + 6

60. 4x2- 5x__

8x2+ 18x - 35

61. 4x - 12_x2

- 25 ÷ 8x - 24_

2x - 10 62. x

2- 5x - 6__

x2- 3x -18

· x2

+ x - 6_x2

- x - 2

Add or subtract. (Lesson 8-4)

63. 2x - 3_x + 1

+ 4x - 9_x - 1

64. 9x_8x - 4

-

10x + 3_12x - 6

65. x2_2x + 7

-

x_x + 2

66. Literature Christopher is reading a book containing 854 pages at a rate of 1.5 pages per minute. Create a table, equation, and graph to represent the number of pages remaining to be read P in relation to time t. (Lesson 9-1)

1

55. an = n3_

3 - 1;

997_3

57. an = -0.05n2 + 0.05n + 0.9; -3.6

1, 2, 4, 8, 16, ...

maximum of 31

x_2x + 7

1_x + 563.

2(3x2 - 5x - 3)__(x + 1)(x - 1)

64. 7x - 6_12(2x - 1)

65. x (x2 - 7)__

(2x + 7)(x + 2)

1

56. an = (-1)n (n2 + n); 110

x - 3_x + 2

868 Chapter 12

In Exercise 50, stu-dents who chose A might be subtract-

ing in the wrong order to find the sequence rule.

In Exercise 52, students who choose C may be starting the sequence with 0 instead of 1. Remind them that the domain is the set of natural numbers, so the first term of a n is a 1 , in which n = 1.

JournalHave students explain how fractals can be used to create patterns. Encourage them to include an exam-ple of a simple fractal.

Have students create rules for two different sequences—one in which the terms increase and one in which they decrease. Then have them find the first 5 terms of each sequence.

12-1


1. a 1 = 4 and a n = 0.5 a n - 1 + 1 for n ≥ 2

4, 3, 2.5, 2.25, 2.125

2. a n = 2 n - 5 -3, -1, 3, 11, 27


3. 20, 40, 80, 160, 320 a n = 10 ( 2 n )

4. 1 _ 2 , 1 _

4 , 1 _

6 , 1 _

8 , 1 _

10 a n = 1 _

2n

5. A ball is dropped and bounces to a height of 6 feet. The ball rebounds to 40% of its previ-ous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 7th bounce?

a n = 6 (0.4) n-1 ;

a 7 = 6 (0.4) 6 ≈ 0.025 ft


Answers 48. Possible answer: No,

a recursive rule is based on one or more previous terms but an explicit rule is based on the term’s n-value.

49. Possible answer: An infinite sequence has ellipsis points, or 3 dots, to indicate that the sequence contin-ues without end. A finite sequence ends and has a last term.

66. t P0 8541 852.52 8513 849.54 8485 846.5

P = 854-1.5t

Geometric Patterns and Tessellations

Sequences of figures can often be described with number patterns.

Polygonal numbers can be represented by dots arranged in the form of polygons. The first four hexagonal numbers are illustrated. The sums show a pattern. The sequence for the total number of dots is 1, 6, 15, 28, ....

Example

The figure shows how regular hexagons can be used to tessellate, or cover, the plane. Write a sequence for the number of hexagons added at each stage. Describe the pattern in the sequence, and find the next term.

Show the number of hexagons added at each stage.

Level 1 2 3 4

Number of Hexagons 1 6 12 18

From the second term on, the number added appears to increase by 6 each time. The next level probably would have 24 hexagons. You can check your conjecture by constructing the next stage of the tessellation and counting the hexagons.

Try This

Write a sequence for the number of polygons added at each stage of the figure. Describe the pattern in the sequence, and find the next term.

1. 2. 3.

Connecting Algebra to Geometry 869

Geometry

See Skills Bank page S66

California StandardsPreparation for 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

1, 3, 5, 7,...; consecutive odd numbers; 9

1, 8, 16, 24,...; after the second term, each term increases by 8; 32.

3, 9, 15, 21, ...; each term increases by 6; 27.

Connecting Algebra to Geometry 869

Geometry

Organizer

Objective: Apply sequences to describe geometric patterns and tessellations.

Online Edition

TeachRememberStudents review and apply geometric patterns.

INTERVENTION For addi-tional review and practice on tessel-lations, see Skills Bank page S66.

CloseAssessHave students create a pattern using an equilateral triangle. Then have them write a number sequence to describe the pattern.

See Skills Bankpage S66

California StandardsPreparation for 22.0


12-2 Series and Summation Notation

ObjectiveEvaluate the sum of a series expressed in sigma notation.

Vocabularyseriespartial sumsummation notation

Why learn this?You can use sums of sequences to find the size of a house of cards. (See Example 4.)

In Lesson 12-1, you learned how to find the nth term of a sequence. Often we are also interested in the sum of a certain number of terms of a sequence. A series is the indicated sum of the terms of a sequence. Some examples are shown in the table.

Sequence 1, 2, 3, 4 2, 4, 6, 8, … 1_2

, 1_3

, 1_4

, 1_5

, 1_6

Series 1 + 2 + 3 + 4 2 + 4 + 6 + 8 + �

1_2

+

1_3

+

1_4

+

1_5

+

1_6

Because many sequences are For the even numbers:

S1 = 2 Sum of first term

S2 = 2 + 4 = 6 Sum of first 2 terms

S3 = 2 + 4 + 6 = 12 Sum of first 3 terms

S4 = 2 + 4 + 6 + 8 = 20 Sum of first 4 terms

infinite and do not have defined sums, we often find partial sums. A partial sum , indicated by Sn,is the sum of a specified number of terms of a sequence.

A series can also be represented by using summation notation , which uses the Greek letter ∑ (capital sigma) to denote the sum of a sequence defined by a rule, as shown.

Last value of k

Explicit formula for sequence

First value of k

1E X A M P L E Using Summation Notation

Write each series in summation notation.

A 3 + 6 + 9 + 12 + 15Find a rule for kth term of the sequence.

a k = 3k Explicit formula

Write the notation for the first 5 terms.

∑

k = 1

5

3k Summationnotation

For sequences with alternating signs:Use (-1)

k+1 if a1 = +.

Use (-1)

k if a1 = -.

B 1_2

-

1_4

+ 1_8

-

1_16

+ 1_32

-

1_64

Find a rule for the kth term.

a k = (-1)

k+1 (

1_2)

k Explicitformula

Write the notation for the first 6 terms.

∑

k = 1

6(-1)

k+1 (

1_2)

k Summationnotation


1a. 2_4

+ 2_9

+ 2_16

+ 2_25

+ 2_36

1b. -2 + 4 - 6 + 8 - 10 + 12


Preparation for 23.0Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

∑

k = 1

5 2_(k + 1)2

∑

k = 1

6(-1)k(2k)

870 Chapter 12

Introduce1

12-2 OrganizerObjective: Evaluate the sum of a series expressed in sigma notation.

Online EditionTutorial Videos

Warm UpFind the first 5 terms of each sequence.

1. a n = 1 _ n 1, 1 _ 2 , 1 _

3 , 1 _

4 , 1 _

5

2. a n = - n 2 -1, -4, -9, -16, -25

3. a n = 2 n 2, 4, 8, 16, 32

4. a n = ( 1 _ 3 )

n - 1 1, 1 _

3 , 1 _

9 , 1 _

27 , 1 _

81

5. a n = n _ n + 1

1 _ 2 , 2 _

3 , 3 _

4 , 4 _

5 , 5 _

6


6. 1, 2, 4, 8, 16, … a n = 2 n - 1

7. 4, 7, 10, 13, 16, …

a n = 3n + 1


Q: Why do mathematicians watch so much TV?

A: They like all types of series.

MotivateAsk students if they have ever received any chain e-mails. Have them imagine that 2 people receive a chain e-mail on Monday, and each of those sends it to 2 people on Tuesday, and so on. How many people have received the e-mail at the end of one week? 254 Explain that the answer to this question is the sum of a sequence, or series.

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.California Standards

Preparation for 23.0

12-2 Series and Summation Notation 871

2E X A M P L E Evaluating a Series

Expand each series and evaluate.

A ∑

k = 3

6 1_ 2 k

∑

k = 3

6 1_ 2 k

= 1_ 2 3

+ 1_ 2 4

+ 1_ 2 5

+ 1_ 2 6

Expand the series by replacing k.

= 1_8

+ 1_16

+ 1_32

+ 1_64

Evaluate powers.

= 8_64

+ 4_64

+ 2_64

+ 1_64

= 15_64

Simplify.

B ∑

k = 1

4(10 - k2)

∑

k = 1

4(10 - k2

) = (10 - 12) + (10 - 22

) + (10 - 32) + (10 - 42

) Expand.

= 9 + 6 + 1 + (-6) Simplify.

= 10


2a. ∑

k = 1

4(2k - 1) 2b. ∑

k = 1

5

-5(2)

k-1

Finding the sum of a series with many terms can be tedious. You can derive formulas for the sums of some common series.

In a constant series, such as 3 + 3 + 3 + 3 + 3, each term has the same value.

The formula for the sum of a constant

series is ∑

k = 1

n

c = nc, as shown.

A linear series is a counting series, such as the sum of the first 10 natural numbers. Examine when the terms are rearranged.

Notice that 5 is half of the number of terms and 11 represents the sum of the first and the last term, 1 + 10. This suggests that the sum of a linear series

is ∑

k = 1

n

k = n__2 (1 + n), which can be written as ∑

k = 1

n

k = n(n + 1)_

2 .

Similar methods will help you find the sum of a quadratic series.

CONSTANT SERIES LINEAR SERIES QUADRATIC SERIES

∑

k = 1

n

c = nc ∑

k = 1

n

k =

n(n + 1)_2

∑

k = 1

n

k2=

n(n + 1)(2n + 1)__6

Summation Formulas

Watch out! A series can have a first value other than k = 1, such as in Example 2A, where k begins at 3.

∑

k = 1

5

3 = 3 + 3 + 3 + 3 + 3 = 5 · 3 = 15

5 terms

⎧

�

⎨

�

⎩

∑

k = 1

n

c = c + c + c + � + c = ncn terms

⎧

�

⎨

�

⎩

∑

1

10

k = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

= (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6)

= 11 + 11 + 11 + 11 + 11 = 5(11) = 55

5 terms

⎧

�

⎨

�

⎩

1 + 3 + 5 + 7 = 16 -5 - 10 - 20 - 40 - 80 = -155

Lesson 12-2 871

Example 1


A. 4 + 8 + 12 + 16 + 20

∑

k = 1

5

4k

B. -1 + 1 _ 4 - 1 _

9 + 1 _

16 - 1 _

25 + 1 _

36

∑

k = 1

6 (-1)

k ( 1 _ k 2

)

Example 2


A. ∑

k = 2

5

1 _ 3 k

1 _ 3 2

+ 1 _ 3 3

+ 1 _ 3 4

+ 1 _ 3 5

= 40 _ 243

B. ∑

k = 1

6 ( k 2 - 10) ( 1 2 -10) +

( 2 2 -10) + ( 3 2 -10) +

( 4 2 -10) + ( 5 2 -10) +

( 6 2 -10) = 31


Additional Examples


EXAMPLE 1

• How do you use the terms of the series to help find the rule for the series?

• How do you know what to write for k in summation notation?

EXAMPLE 2

• How do you know the starting and ending values when you expand and evaluate a series?

• How does expanding a series differ from evaluating the series?

Reading Math Use words to help students read sum-mation notation; for

example, ∑

k = 1

5

2k

represents “the sum from 1 to 5 of 2k.”

Guided InstructionDefine a series and explain the relation-ship to a sequence. Review summation notation, and ask students to clarify what each variable or expression represents. Reinforce the notation verbally and visually because this material may be new for most students. Point out that the summation formulas on p. 871 are convenient ways to find the sums of common series.

Teach2

Through Cooperative Learning

Have groups of students create a sequence and a related series to model a real-world situation. Have them generate a question that draws on the terms of a sequence and a question that requires the sum of the related series. For example, cans are stacked in a pyramidal display with 1 in the top row, 2 in the 2nd row, 3 in the 3rd row, etc. How many cans are in the 5th row? sequence How many total cans are in a display with 10 rows? series



3E X A M P L E Using Summation Formulas

Evaluate each series.

A ∑

k = 5

10

8 Constant series

Method 1 Use the summation Method 2 Expand and evaluate. formula.

∑

k = 5

10

8 = 8 + 8 + 8 + 8 + 8 + 8 = 48 There are 6 terms. 6 terms

∑

k = 5

10

8 = nc = 6(8) = 48

B ∑

k = 1

5

k Linear series

Method 1 Use the summation Method 2 Expand and evaluate. formula.

∑

k = 1

5

k = n(n + 1)_

2 =

5(6)_2

= 15 ∑

k = 1

5

k = 1 + 2 + 3 + 4 + 5 = 15

C ∑

k = 1

7

k2 Quadratic series

Method 1 Use the summation Method 2 Use a graphing formula. calculator.

∑

k = 1

7

k2=

n(n + 1)(2n + 1)__6

= 7(7 + 1)(2 · 7 + 1)__

6

= 56(15)_

6

= 140


3a. ∑

k = 1

60

4 3b. ∑

k = 1

15

k 3c. ∑k = 1

10

k2

4E X A M P L E Problem-Solving Application

Ricky is building a card house similar to the one shown. He wants the house to have as many stories as possible with a deck of 52 playing cards. How many stories will Ricky’s house have?

1 Understand the Problem

The answer will be the number of stories, or rows, in the card house.List the important information:• He has 52 playing cards.• The house should have as many stories as possible.

2 Make a Plan

Make a diagram of the house to better understand the problem. Find a pattern for the number of cards in each story. Write and evaluate the series.

When counting the number of terms, you must include both the first and the last. For example,

∑

k = 5

10

8 has six terms,

not five.k = 5, 6, 7, 8, 9, 10

⎧

�

⎨

�

⎩

240 120 385

872 Chapter 12

Example 3


A. ∑

k = 4

10 6 42

B. ∑

k = 1

8 k 36

C. ∑

k = 1

12 k 2 650

Example 4

Sam is laying out patio stones in a triangular pattern. The first row has 2 stones and each row has 2 additional stones, as shown below. How many complete rows can he make with a box of 144 stones? 11


Additional Examples


EXAMPLE 3

• How do you recognize a constant, linear, or quadratic series?

• Are there types of series other than constant, linear, or quadratic? How do you know?

EXAMPLE 4

• How does a table or diagram help you identify a pattern?

• Why do you need to identify a pat-tern at all?

• How does the series represent the total?

Through Visual Cues

Have students create a poster of summa-tion notation to display in the classroom. Have them use color coding to indicate that k is the value that changes with each new term in a series.


3 Solve

Make a table and a diagram.

Row 1 2 3 4

Diagram

Cards 2 5 8 11

The number of cards increases by 3 in each row. Write a series to represent the total number of cards in n rows.

∑

k = 1

n

(3k - 1) , where k is the row number and n is the total number of rows

Evaluate the series for several n-values.

∑

k = 1

4(3k - 1) = ⎡

⎣ 3(1) - 1⎤

⎦ +

⎡

⎣ 3(2) - 1⎤

⎦ +

⎡

⎣ 3(3) - 1⎤

⎦ +

⎡

⎣ 3(4) - 1⎤

⎦

= 26

∑

k = 1

5(3k - 1) =

⎡

⎣ 3(1) - 1⎤

⎦ +

⎡

⎣ 3(2) - 1⎤

⎦ +

⎡

⎣ 3(3) - 1⎤

⎦ +

⎡

⎣ 3(4) - 1⎤

⎦ +

⎡

⎣ 3(5) - 1⎤

⎦

= 40

∑

k = 1

6(3k - 1) = ⎡

⎣ 3(1) - 1⎤

⎦ +

⎡

⎣ 3(2) - 1⎤

⎦ +

⎡

⎣ 3(3) - 1⎤

⎦ +

⎡

⎣ 3(4) - 1⎤

⎦ +

⎡

⎣ 3(5) - 1⎤

⎦ +

⎡

⎣ 3(6) - 1⎤

⎦

= 57

Because Ricky has only 52 cards, the house can have at most 5 stories.

4 Look Back

Use the table to continue the pattern. The 5th row would have 14 cards. S5 = 2 + 5 + 8 + 11 + 14 = 40. The next row would have more than 12 cards, so the total would be more than 52.

4. A flexible garden hose is coiled for storage. Each subsequent loop is 6 inches longer than the preceding loop, and the innermost loop is 34 inches long. If there are 6 loops, how long is the hose?

THINK AND DISCUSS 1. Explain the difference between a sequence and a series.

2. Explain what each of the variables represents in the notation ∑

k = m

n

k.

3. GET ORGANIZED Copy and complete the graphic organizer. Write the general notation and an example for each term.

294 in., or 24 1_2 ft

Lesson 12-2 873

Students may attempt to find the partial sum of a series by evaluat-ing the expression for only the final value of k. Stress that the notation means sum, so students should be adding at least two terms to evalu-ate a series.

Technology Students are taught to use the sum( function on graphing calcu-

lators in the Technology Lab follow-ing Lesson 12-2. You may want to teach that lab before or during this lesson.

Statistics The Greek letter

Σ (capital sigma) denotes the sum of a sequence

defined by a rule. The Greek letter σ (lower case sigma) denotes the standard deviation of a set of num-bers.





and INTERVENTIONSummarizeDisplay an expression in summation

notation, such as ∑

k = 3

6 (5k - 4) . Ask

students to describe how they would eval-uate this sum, taking care to discuss each step. Have a volunteer complete each step as it is discussed. Possible answer: Identify the values of k that will be included: 3, 4, 5, and 6. Evaluate the expression for each of these terms and then find the sum: 11 + 16 + 21 + 26 = 74.

Close3 Answers to Think and DiscussPossible answers:

1. A sequence is an ordered list of terms. When the terms are added together, a series is created. The series is an indi-cated sum of the sequence.

2. m is the first replacement for k, n is the last replacement for k, and k is the explicit formula.

3. See p. A13.


ExercisesExercises

GUIDED PRACTICE 1. Vocabulary Give an example of summation notation.



2. 1 + 1_4

+ 1_9

+ 1_16

+ 1_25

3. -3 + 6 - 9 + 12 -15

4. 1 + 10 + 100 + 1000 + 10,000 5. 100 + 95 + 90 + 85 + 80



6. ∑

k = 1

5

k3 7. ∑

k = 1

4

(-1)

k+1 12_k2

8. ∑

k = 5

10

-5k



9. ∑

k = 1

21

k 10. ∑

k = 1

20

k2 11. ∑

k = 15

35

6


12. Finance Melinda makes monthly car payments of $285 each month. How much will she have paid after 2 years? 5 years?

PRACTICE AND PROBLEM SOLVING


13–16 1 17–19 2 20–22 3 23 4

Independent Practice Write each series in summation notation.

13. 1.1 + 2.2 + 3.3 + 4.4 + 5.5 14. 1_2

+ 2_3

+ 3_4

+ 4_5

+ 5_6

15. 11 - 12 + 13 - 14 + 15 - 16 16. 1 + 2 + 4 + 8 + 16 + 32


17. ∑

k = 1

5⎡

⎣ 8(k + 1)⎤

⎦ 18. ∑

k = 2

7

(-2)

k 19. ∑

k = 1

4 k - 1_k + 1


20. ∑

k = 1

99

k 21. ∑

k = 11

88

2.5 22. ∑

k = 1

25

k2

23. Retail A display of soup cans is arranged with 1 can on top and each row having an additional can. How many cans are in a display of 20 rows?


24. -1 + 4 - 9 + 16 - 25 + 36 25. 25 + 24 + 23 + � + 1

26. 1_3

+ 1_9

+ 1_27

+ 1_81

+ 1_243

27. -800 - 80 - 8 - 0.8 - 0.08

28. 10.8 + 10.5 + 10.2 + 9.9 29. 9 - 16 + 25 - 36 + 49 - 64

30. -3.9 + 4.4 - 4.9 + 5.4 - 5.9 31. 0 + 3.4 + 6.8 + 10.2 + 13.6

32. 3 + 3_2

+ 1 + 3_4

+ 3_5

33. 1000 + 100 + 10 + 1 + 1_10

12-2KEYWORD: MB7 12-2

KEYWORD: MB7 Parent


Extra Practice

California StandardsPrep for 23.0; 1.0

34. Travel The distance from St. Louis, Missouri, to Los Angeles, California, is 1596 miles. Michael plans to travel half the distance on the first day and half the remaining distance each day after that. Write a series in summation notation for the total distance he will travel in 5 days. How far will Michael travel in the 5 days?

∑

k = 1

n

k

∑

k = 1

5

( 1_k2 ) ∑

k = 1

6(-1)k(3k)

∑

k = 1

5(10)k-1

1 + 8 + 27 + 64 + 125 = 225 12 - 3 + 4_3 -

3_4 = 9 7_

12231 2870 126

$6840; $17,100

∑

k = 1

51.1k ∑

k = 1

5 k_k + 1

15. ∑

k = 1

6(-1)k+1(k + 10)

16 + 24 + 32 + 40 + 48 = 160 0 + 1_3

+ 2_4

+ 3_5

= 43_30

4950 195 5525

210 cans

∑

k = 1

6(-1)k k 2

∑

k = 1

5

( 1_3 )

k

∑

k = 1

5⎡ ⎣ 100 - 5(k - 1)⎤

⎦

∑

k = 1

5 3_k

∑

k = 1

25(26 - k)

∑

k = 1

51596 ( 1_

2 )k; 1546 1_

8 mi

∑

k = 1

5 1000_ 10 k-1

∑

k = 1

6 2 k-1

874 Chapter 12

ExercisesExercises

12-2 PRACTICE A 12-2 PRACTICE B

Assignment Guide


If you finished Examples 1–2 Basic 13–19, 24–26, 36–39 Proficient 13–19, 24–29, 36–39 Advanced 13–19, 24–33, 36–39


62–68 Proficient 13–29, 34–46, 48–58,

62–68 Advanced 14–34 even, 35–68

Homework Quick CheckQuickly check key concepts.Exercises: 16, 18, 20, 30, 34

Answers 8. -25 - 30 - 35 - 40 - 45 - 50

= -225

18. 4 - 8 + 16 - 32 + 64 - 128 = - 84

27. ∑

k = 1

5

- 800 ( 1_10 )

k-1

28. ∑

k = 1

4

11.1 - 0.3k

29. ∑

k = 1

6

(-1)k+1(k + 2)2

30. ∑

k = 1

5

(-1)k(3.4 + 0.5k)

31. ∑

k = 1

5

3.4k - 3.4 , or ∑k = 1

5

3.4 (k - 1)

12-2

20 ft 20 ft


35. Safety An employer uses a telephone tree to notify employees in the case of an emergency closing. When the office manager makes the decision to close, she calls 3 people. Each of these people calls 3 other people, and so on.

a. Make a tree diagram with 3 levels to represent the problem situation.

b. Write and evaluate a series to find the total number of people notified after 5 levels of calls.

c. What if...? Suppose that the phone tree involves calling 5 people at each level. How many more people would be notified after 5 levels of calls?


36. ∑

k = 1

6

(k2+ 1) 37. ∑

k = 1

6(-1)

k 5k 38. ∑

k = 3

6 1_2k

39. ∑

k = 1

6(3k - 2) 40. ∑

k = 6

11

12(k - 2) 41. ∑

k = 1

5 k2_5k

42. Architecture A hotel is being built in the shape of a pyramid, as shown in the diagram. Each square floor is 10 feet longer and 10 feet wider than the floor above it.

a. Write a series that represents the total area of n floors of the hotel.

b. How many stories must the hotel be to have at least 50,000 square feet of floor area?

Estimation Use mental math to estimate each sum. Then compare your answer to the sum obtained by using a calculator.

43. 10 + 11 + 12 + � + 29 + 30

44. 1 + 3 + 5 + � + 97 + 99

45. -2 + (-4) + (-6) + � + (-98) + (-100)

46. Physics The distance that an object Distance Fallenin Each Time Interval

TimeInterval

Leonardo’sRule

Galileo’sRule

12345

12345

13579

falls in equal time intervals is represented in the table. Rules created by Leonardo da Vinci and Galileo are shown. (In this general case, specific units do not apply to time or distance.)

a. Write the series for each rule for 5 intervals, and find the respective sums. What does the sum of the series for 5 intervals represent?

b. Write each series in summation notation. Then evaluate each series for n = 10.

c. By the current rule, the distance fallen in each interval is 1, 4, 9, 16, 25…. How do Leonardo’s rule and Galileo’s rule compare to the current rule?

47. Critical Thinking Some mathematical properties may be applied to series.

a. Evaluate ∑

k = 1

10

3n and 3 ∑

k = 1

10

n. Make a conjecture based on your answer.

b. Evaluate ∑

k = 1

10

n + ∑

k = 1

10

2 and ∑

k = 1

10

(n + 2) . Make a conjecture based on your answer.

The

Gra

nger

Col

lect

ion,

New

Yor

k

At the age of 10, German mathematician Carl Friedrich Gauss discovered a quick method for adding the first 100 natural numbers. His method gave us the summation formula for a linear series.

Math History

3542

42a. ∑

k = 1

n

100(k + 1)2 10 stories

2500

-2550

420

Lesson 12-2 875

Students may have trouble with series in which the terms have alter-nating signs, such as Exercises 24, 29, and 30. Remind them that each of these must contain either (-1) k if the first term is negative or (-1) k + 1 if the first term is positive.

Critical Thinking After students complete Exercises 47 and 48, have

them consider whether the addition of series is commutative or associative.

Answers 35a.

b. ∑

k = 1

5

3 k = 3 + 9 + 27 + 81

+ 243 = 363

36. 2 + 5 + 10 + 17 + 26 + 37 = 97

37. -5 + 10 - 15 + 20 - 25 + 30 = 15

38. 1_6

+1_8

+1_10

+1_12

=19_40

39. 1 + 4 + 7 + 10 + 13 + 16 = 51

40. 48 + 60 + 72 + 84 + 96 + 108 = 468

41. 1_5

+2_5

+3_5

+4_5

+ 1 = 3

46a. Leonardo: 1 + 2 + 3 + 4 + 5 = 15; Galileo: 1 + 3 + 5 + 7 + 9 = 25; the total distance traveled in 5 equal units of time

b. Leonardo: ∑k = 1

n

k; 55;

Galileo: ∑k = 1

n

(2k - 1) ; 100

c. Objects fall farther under the current rule.

47a. Both equal 165;

∑

k = 1

n

ca k = c∑

k = 1

n

a k.

b. Both equal 75;

∑

k = 1

n

(a k + b k)

= ∑

k = 1

n

a k + ∑

k = 1

n

b k.

12-2 PRACTICE C


49. Multi-Step Examine the pattern made by toothpick squares with increasing side lengths.

a. Write a sequence for the number of toothpicks added to form each new square.

b. Write and evaluate a series in summation notation to represent the total number of toothpicks in a square with a side length of 6 toothpicks.

50. Critical Thinking Are the sums of 1 + 3 + 5 + 7 + 9 and 9 + 7 + 5 + 3 + 1 the same? Do these series have the same summation notation? Explain.

51. Write About It Explain why Sn represents a partial sum and not a complete sum of the terms of a sequence.

52. Which notation accurately reflects the series ∑

k = 1

7(-1)

k(3k)?

3 + 6 + 9 + 12 + 15 + 18 + 21 -3 + 6 - 9 + 12 - 15 + 18 - 21

3 - 6 + 9 - 12 + 15 - 18 + 21 -3 - 6 - 9 - 12 - 15 - 18 - 21

53. Which notation accurately reflects the series 1_2

+ 1_4

+ 1_6

+ 1_8

?

∑

k = 1

4 k_2

∑

k = 1

4 1_2k

∑

k = 1

4 1_2k

∑

k = 1

4 1_k + 2

54. What is the value of ∑

k = 1

6

k2?

36 55 91 273

55. Find the sum of the series 1_3

+ 1_6

+ 1_12

+ 1_24

.

1_45

4_45

7_12

5_8


The series ∑

k = 1

n

(3k2+ 3k) gives the total number of sticks needed

to make a tetrahedral kite with n layers.

a. Expand and evaluate the series to find out how many sticks are needed to make a kite with 5 layers.

b. Use the properties

∑

k = 1

n

(ak + bk) = ∑

k = 1

n

ak + ∑

k = 1

n

bk and ∑

k = 1

n

c a k = c ∑

k = 1

n

ak

to rewrite the series as a multiple of a quadratic series plus a multiple of a linear series.

c. Use summation formulas to determine how many sticks are needed to make a kite with 17 layers. 5814

49b. ∑

k = 1

64k ; 84

a n = 4n

876 Chapter 12

Exercise 48 involves expanding and evaluating the partial

sum of a series that models the lay-ers of a tetrahedral kite. This exercise prepares students for the Concept Connection on page 888.

In Exercise 52, stu-dents who chose A or D probably did not

recognize the effect that (-1) k has on the terms of the series. Students who chose B may be confusing the effect of odd and even exponents in that expression. Review how to evaluate (-1) k for odd and even values of k.

In Exercise 53, students may be confused by the fractions as terms of the series. Because all the numerators are 1, encourage them to focus on the denominators. They can find a rule for the denominators and then write it as a denominator under 1.

Answers 48a. 6 + 18 + 36 + 60 + 90 = 210

b. 3 ∑

k = 1

n

k 2 + 3 ∑

k = 1

n

k

50. Yes, both series have the same sum, 25. However, their summation notations are not the same because

∑

k = 1

5 (2k - 1) = 1 + 3 + 5 + 7 + 9

and

∑

k = 1

5 (11 - 2k) = 9 + 7 + 5 + 3 + 1 .

51. Possible answer: S n represents a partial sum because it is the sum of only a specified number of terms. Many series are infinite and do not have actual sums.

56. Possible answer: A sequence would give the number of cans in the 20th row because each term in the sequence represents the number of cans in a row. A series would give the total num-ber of cans in the stack.

59–61. See p. A46.

62–64. For graphs, see p. A46.



56. Short Response The number of cans in each row of a pyramidal stack is 1, 4, 9, 16, .... Would a sequence or series be used to find the number of cans in the 20th row? Explain.

CHALLENGE AND EXTENDWrite each series in summation notation. Then find the sum.

57. 1 + 2 + 3 + � + 1000 58. 1 2 + 2 2 + 3 2 + � + 2 5 2

Prove each summation property for the sequences ak and bk.

59. ∑

k = 1

n

c a k = c∑

k = 1

n

ak 60. ∑

k = 1

n

(ak + bk) = ∑

k = 1

n

ak + ∑

k = 1

n

bk

61. Critical Thinking What might the sum of the sequence 1 - 1 + 1 - 1 + 1 - 1 + �

be if it continues forever? Explain.

SPIRAL REVIEWFind the intercepts of each line, and graph the lines. (Lesson 2-3)

62. 3x - 4y = 12 63. -6x + 3y = -18 64. 10x + 15y = -5

65. Architecture The height in feet of an elevator above ground is modeled by h(t) = 8 ⎪t - 6⎥ + 10, where t is the time in seconds. What is the minimum height of the elevator? (Lesson 2-9)

Find the first 5 terms of each sequence. (Lesson 12-1)

66. an = (

1_2

n + 2) 2

67. a1 = 2, an = (an-1)

2- 1 68. an = 4 n_

2

Q: What high school math classes did you take?

A: Algebra 1, Geometry, Business Math, and Algebra 2

Q: Why did you decide to pursue a career in nursing?

A: I’ve always liked science, especially biology and anatomy. I also wanted a career where I can help people, so a job in medicine seemed perfect.

Q: How is math used in nursing?

A: Nurses have to make a lot of calculations to ensure that patients receive the correct dosages of medicine. They also use complicated instruments to measure and monitor different body functions.

Q: What are your future plans?

A: I’ll finish my associate’s degree in nursing at the end of the semester. I hope to get a job at one of the local hospitals.

Steven HoweNursing student

KEYWORD: MB7 Career

10 ft

25_4 , 9, 49_

4 , 16, 81_

42, 3, 8, 63, 3968 2, 8, 32, 128, 512

62. x-int.: 4; y-int.: -3

63. x-int.: 3; y-int.: -6

64. x-int.: - 1_2 ;

y-int.: - 1_3

∑

k = 1

1000k ; 500,500 ∑

k = 1

25k2 ; 5525

Lesson 12-2 877


In Exercise 48b, some students might have difficulty recognizing how the properties correspond

to the series ∑

k = 1

n

(3 k 2 + 3k) . Point

out that a k = 3 k 2 and b k = 3k, so

∑k = 1

n

(3 k 2 + 3k) = 3 ∑

k = 1

n

k 2 + 3 ∑

k = 1

n

k .

Now students can use the formulas to evaluate each part for n = 17 in Exercise 48c.

JournalHave students explain summation notation using words and by provid-ing an example. Encourage them to include an example of a partial sum as well.

Have students create two different series using summation notation. One should be a series with alternat-ing signs. Have students expand and evaluate each series for several dif-ferent values of k and n.

12-2


1. 1 - 10 + 100 - 1000 + 10,000

2. 1 _ 5 + 1 _

10 + 1 _

15 + 1 _

20 + 1 _

25

1. ∑

k = 1

5

(-10) k - 1 2. ∑

k = 1

5

1 _ 5k


3. ∑

k = 4

8 (2k - 1) 55 4. ∑

k = 10

25 4 64

5. ∑

k = 1

25 k 325 6. ∑

k = 1

9 k 2 285

7. Ann is making a display of hand-held computer games. There will be 1 game on top. Each row will have 8 additional games. She wants the display to have as many rows as possible with 100 games. How many rows will Ann’s display have? 5



Evaluate Sequences and SeriesGraphing calculators have built-in features that help you generate the terms of a sequence and find the sums of series.

Activity

Use a graphing calculator to find the first 7 terms of the sequence an = 1.5n + 4. Then find the sum of those terms.

1 Find the first 7 terms of the sequence.

Enter the LIST operations menu by pressing and scrolling right to the OPS menu. Then select the sequence command 5:seq(.

The sequence command takes the following four expressions separated by commas.

Enter the rule, using x as the variable. Enter 1 for the starting term and 7 for the ending term. Close the parentheses, and then press .

The terms of the sequence will be displayed in brackets. Use the arrow keys to scroll to see the rest of the terms.

The first 7 terms are 5.5, 7, 8.5, 10, 11.5, 13, and 14.5.

2 Find the sum of the terms.

Enter the LIST math menu by pressing ,and scrolling right to the MATH menu. Then select the sum command 5:sum(.

Follow the steps for entering a sequence as shown in Step 1.

The sum of the first 7 terms is 70.

Try This

Find the first 8 terms of each sequence. Then find the sum of those 8 terms.

1. an = 2 n2- 5 2. an =

1_4

(2)

n-1 3. an = 0.3n + 1.6

4. an = 20n 5. an = n3- 2n 6. an = 0.1 (5)

n

7. Critical Thinking Find the next 5 terms of the sequence 200, 182, 164, 146, 128, .... Then find the sum of those 5 terms.

12-2

Use with Lesson 12-2 California StandardsPreparation for 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.Also covered: Preparation for 23.0

Organizer

Objective: Use a graphing calculator to generate the terms of a sequence and find the sums of series.

Materials: graphing calculator

Online EditionTechKeys

ResourcesTechnology Lab Activities

12-2 Lab Recording Sheet

TeachDiscussHave students use seq( to find the first 5 terms of a n = 2 n 2 . Discuss the expressions that must be entered. Then have them use sum( to find the sum of the terms. Discuss why the sum is evaluated using sum(seq(.

CloseKey ConceptTo evaluate the partial sum of a series by using the sum( and seq( functions on a graphing calculator

AssessmentJournal Have students explain how to use a graphing calculator to evalu-ate a sequence or a series.

Use with Lesson 12-2

Answers to Try This 1. -3, 3, 13, 27, 45, 67, 93, 123; 368

2. 0.25, 0.5, 1, 2, 4, 8, 16, 32; 63.75

3. 1.9, 2.2, 2.5, 2.8, 3.1, 3.4, 3.7, 4; 23.6

4. 20, 40, 60, 80, 100, 120, 140, 160; 720

5. -1, 4, 21, 56, 115, 204, 329, 496; 1224

6. 0.5, 2.5, 12.5, 62.5, 312.5, 1562.5, 7812.5, 39,062.5; 48,828

7. 110, 92, 74, 56, 38; 370

878 Chapter 12

California StandardsPreparation for 22.0Also covered: Preparation for 23.0Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

2.00

1.50

1.00

0.50

Weight (oz)

2005 U.S. Postage Costs

2 4 6 8

Co

st (

$)

0

12-3 Arithmetic Sequences and Series 879

12-3 Arithmetic Sequences and Series

ObjectivesFind the indicated terms of an arithmetic sequence.

Find the sums of arithmetic series.

Vocabularyarithmetic sequencearithmetic series

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Who uses this?You can use arithmetic sequences to predict the cost of mailing letters.

The cost of mailing a letter in 2005 gives the sequence 0.37,0.60,0.83,1.06, ….This sequence is called an arithmetic sequence because its successive terms differ by the same number d (d ≠ 0) , called the common difference. For the mail costs, d is 0.23, as shown.

Term a1 a2 a3 a4

Value 0.37 0.60 0.83 1.06

Differences 0.23 0.23 0.23

Recall that linear functions have a constant first difference. Notice also that when you graph the ordered pairs (n, an) of an arithmetic sequence, the points lie on a straight line. Thus, you can think of an arithmetic sequence as a linear function with sequential natural numbers as the domain.

1E X A M P L E Identifying Arithmetic Sequences

Determine whether each sequence could be arithmetic. If so, find the common first difference and the next term.

A -3, 2, 7, 12, 17, …

-3, 2, 7, 12, 17Differences 5 5 5 5The sequence could be arithmetic with a common difference of 5. The next term is 17 + 5 = 22.

B -4, -12, -24, -40, -60, …

-4, -12, -24, -40, -60Differences -8 -12 -16 -20The sequence is not arithmetic because the first differences are not common.

Determine whether each sequence could be arithmetic. If so, find the common difference and the next term.

1a. 1.9, 1.2, 0.5, -0.2, -0.9, ... 1b. 11_2

, 11_3

, 11_4

, 11_5

, 11_6

, ...

Each term in an arithmetic sequence is the sum of the previous term and the common difference. This gives the recursive rule an = an-1 + d. You also can develop an explicit rule for an arithmetic sequence.


23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.Also covered: 22.0

arithmetic; -0.7; -1.6 not arithmetic

Lesson 12-3 879

Introduce1

Objectives: Find the indicated terms of an arithmetic sequence.

Find the sums of arithmetic series.

Online EditionTutorial Videos, Interactivity, TechKeys

Warm UpFind the 5th term of each sequence.

1. a n = n + 6 11

2. a n = 4 - n -1

3. a n = 3n + 4 19


4. 4, 5, 6, 7, 8, … a n = n + 3

5. -3, -1, 1, 3, 5, … a n = 2n - 5

6. 3 _ 2 , 2, 5 _

2 , 3, 7 _

2 , … a n = n _

2 + 1


Teacher: You have the terms of the sequence all mixed up.

Student: If you wanted them in order, why didn’t you just say so?

MotivatePose this famous question to your students: “What is the sum of the first 100 natural num-bers?” 5050 Explain that the sum is found by adding 50 pairs of numbers (1 + 100, 2 + 99, 3 + 98, …) , each with a sum of 101. Explain that this method, discovered by a young Carl Friedrich Gauss (1777–1855), works for arithmetic sequences.

12-3 Organizer

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.

California Standards 23.0

Also covered: 22.0Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.


Notice the pattern in the table. Each term is the sum of the first term and a multiple of the common difference.

This pattern can be generalized into a rule for all arithmetic sequences.

The nth term an of an arithmetic sequence is given by

an = a1 + (n - 1)d

where a1 is the first term and d is the common difference.

General Rule for Arithmetic Sequences

2E X A M P L E Finding the nth Term Given an Arithmetic Sequence

Find the 10th term of the arithmetic sequence 32, 25, 18, 11, 4, … .

Step 1 Find the common difference: d = 25 - 32 = -7.

Step 2 Evaluate by using the formula.

a n = a1 + (n - 1)d General rule

a 10 = 32 + (10 - 1)(-7) Substitute 32 for a1, 10 for n, and -7 for d.

= -31 Simplify.

The 10th term is -31.

Check Continue the sequence.

n 1 2 3 4 5 6 7 8 9 10

an 32 25 18 11 4 -3 -10 -17 -24 -31

Find the 11th term of each arithmetic sequence.

2a. -3, -5, -7, -9, … 2b. 9.2, 9.15, 9.1, 9.05, …

Postage Costs per Ounce

n an

1 a1 = 0.37 + 0(0.23)

2 a2 = 0.37 + 1(0.23)

3 a3 = 0.37 + 2(0.23)

4 a4 = 0.37 + 3(0.23)

n an = 0.37 + (n - 1)(0.23)

✔

I like to check the value of a term by using a graphing calculator.

I enter the function for the nth, or general, term. For Example 2A, enter y = 32 + (x - 1)(-7).

I then use the table feature. Start at 1 (for n = 1),and use a step of 1. Then find the desired term (y-value) as shown for n = 10.

Finding the nth Term

Diana WatsonBowie High School

-23 8.7

880 Chapter 12

Example 1

Determine whether each sequence could be arithmetic. If so, find the common first difference and the next term.

A. -10, -4, 2, 8, 14, … yes; 6; 20

B. -2, -5, -11, -20, -32, … no

Example 2

Find the 12th term of the arith-metic sequence

20, 14, 8, 2, -4, … . -46


Additional Examples


EXAMPLE 1

• Why do you have to find the first differences?

• How could you use the graph of a sequence to decide whether the sequence is arithmetic?

EXAMPLE 2

• What must you know to be able to find the nth term of a sequence? Explain.

Technology To display the terms of an arithmetic sequence on a graphing

calculator, enter the first term, press

, press , enter the com-

mon difference, and press repeatedly.

Guided InstructionIntroduce an arithmetic sequence such as 4, 7, 10, 13, … . Make the connection with linear functions that students have already studied. Show students how the general rule for an arithmetic sequence can be used in a number of ways as you review Examples 2, 3, and 4. When introducing the sum formula for arithmetic series, you may want to remind students of the sum of a linear series (Lesson 12-2).

Teach2

Through Modeling

Separate students into small groups and give each group a set of small manipulatives, such as centimeter squares or cubes (MK). Have students create patterns by adding a specified number of items to each row. As they construct the model, have them write down the number of items in each row and the cumulative total for each row. Then have them write a rule for the sequence that describes their pattern.


3E X A M P L E Finding Missing Terms

Find the missing terms in the arithmetic sequence 11, , , , -17.

Step 1 Find the common difference.

a n = a1 + (n - 1)d General rule

-17 = 11 + (5 - 1)d Substitute -17 for an, 11 for a1, and 5 for n.

-7 = d Solve for d.

Step 2 Find the missing terms using d = -7 and a1 = 11.

a2 = 11 + (2 - 1)(-7) a3 = 11 + (3 - 1)(-7) a4 = 11 + (4 - 1)(-7)

= 4 = -3 = -10

The missing terms are 4, -3, and -10.

3. Find the missing terms in the arithmetic sequence 2, , , , 0.

Because arithmetic sequences have a common difference, you can use any two terms to find the difference.

4E X A M P L E Finding the n th Term Given Two Terms

Find the 6th term of the arithmetic sequence with a9 = 120 and a14 = 195.

Step 1 Find the common difference.

an = a1 + (n - 1)d

a14 = a9 + (14 - 9)d Let an = a14 and a1 = a9. Replace 1 with 9.

a14 = a9 + 5d Simplify.

195 = 120 + 5d Substitute 195 for a14 and 120 for a9.

75 = 5d

15 = d

Step 2 Find a1 .

an = a1 + (n - 1)d General rule

120 = a1 + (9 - 1)(15) Substitute 120 for a9, 9 for n, and 15 for d.

120 = a1 + 120 Simplify.

0 = a1

Step 3 Write a rule for the sequence, and evaluate to find a6 .


an = 0 + (n - 1)(15) Substitute 0 for a1 and 15 for d.

a6 = 0 + (6 - 1) 15 Evaluate for n = 6.

= 75

The 6th term is 75.


4a. a2 = -133 and a3 = -121 4b. a3 = 20.5 and a8 = 13

3_2 , 1, 1_

2

-258.5

Lesson 12-3 881

Students sometimes confuse a nand a 1 or n and a n. Emphasize the importance of writing the rule as the first step in using it to find missing terms. Discuss how n is different from a n.

Example 3

Find the missing terms in the arithmetic sequence

17, , , , -7. 11, 5, -1

Example 4

Find the 5th term of the arithme-tic sequence with a 8 = 85 and a 14 = 157. 49


Additional Examples


EXAMPLE 3

• What is the first step in finding the common difference? Explain.

• After you find the common differ-ence, is there a way to find the missing terms other than using the rule for the nth term? Explain.

EXAMPLE 4

• Why can you replace a n and a 1 with other terms from the sequence?

Visual It often helps to visually represent the sequences with missing

terms. For Example 4, you may want to show something like this.

120 195

a 9 a 10 a 11 a 12 a 13 a 14

I tell students that a n = a 1 + (n - 1) d works with any two terms. For Example 4:

a 14 = a 9 + (14 - 9) d

195 = 120 + 5d

75 = 5d

15 = d

Then solve for a 6 by using a 9 = a 6 + (9 - 6)d, where a 9 = 120 and d = 15. In a few steps, you find that a 6 = 75. You don’t ever need to find a 1 . Students really like this and it always works.

Dave BarkerLos Alamitos, CA

Teacher to Teacher


In Lesson 12-2 you wrote and evaluated series. An arithmetic series is the indicated sum of the terms of an arithmetic sequence. You can derive a general formula for the sum of an arithmetic series by writing the series in forward and reverse order and adding the results.

WORDS NUMBERS ALGEBRA

The sum of the first nterms of an arithmetic series is the product of the number of terms and the average of the first and last terms.

The sum of 2 + 4 + 6 + 8 + 10 is

5(

2 + 10_2 ) = 5 (6) = 30.

Sn = n(

a1 + an_2 ),

where n is the number of terms, a1 is the first term, and an is the nth term.

Sum of the First Terms of an Arithmetic Series

5E X A M P L E Finding the Sum of an Arithmetic Series

Find the indicated sum for each arithmetic series.

A S 15 for 25 + 12 + (-1) + (-14) +

Find the common difference.d = 12 - 25 = -13

Find the 15th term.a15 = 25 + (15 - 1) (-13)

= -157Find S15 .

S n = n(

a1 + an_2 ) Sum formula

S 15 = 15(

25 + (-157)__2 ) Substitute.

= 15 (-66) = -990

Check Use a graphing calculator.

✔

S n = a1 + (a1 + d) + (a1 + 2d ) + … + an

S n = an + (an - d) + (an - 2d ) + … + a1

2Sn = (a1 + an) + (a1 + an) + (a1 + an) … + (a1 + an)

(a1 + an) is added n times

2 Sn = n(a1 + an)

S n = n(a1 + an)_

2, or Sn = n(

a1 + an_2 )

⎧

�

⎨

�

⎩

These sums are actually partial sums.You cannot find the complete sum of an infinite arithmetic series because the term values increase or decrease indefinitely.


5a. S16 for 12 + 7 + 2 + (-3) + � 5b. ∑

k = 1

15(50 - 20k)

B ∑

k = 1

12

(3 + 4k)

Find the 1st and 12th terms. a 1 = 3 + 4 (1) = 7 a 12 = 3 + 4(12) = 51

Find S12 .

S n = n(

a1 + an_2 )

S 12 = 12(

7 + 51_2 )

= 348


✔

-408 -1650

882 Chapter 12

Math Background You may wish to show some additional numerical exam-

ples before the formal derivation of the sum of the first n terms of an arithmetic series. For example:

1 + 2 + 3 + 4 = 4 (1 + 4)

_ 2 ; and

so on.

Example 5


A. S 18 for 13 + 2 + (-9) + (-20) + … -1449

B. ∑

k = 1

15 (5 + 2k) 315

Example 6

The center section of a concert hall has 15 seats in the first row and 2 additional seats in each subsequent row.

A. How many seats are in the 20th row? 53

B. How many seats in total are in the first 20 rows? 680


Additional Examples


EXAMPLE 5

• How do you know which term(s) to find to evaluate the sum?

• When do you find the common dif-ference to find the sum? Explain.

EXAMPLE 6

• Why do you use an arithmetic sequence to solve the problem?

• How do you know which part of the problem is solved using an arithmetic sequence and which is solved using an arithmetic series?


6E X A M P L E Theater Application

The number of seats in the first 14 rows of the center orchestra aisle of the Marquis Theater on Broadway in New York City form an arithmetric sequence as shown.

A How many seats are in the 14th row?Write a general rule using a1 = 11 and d = 1.

a n = a1 + (n - 1)d Explicit rule for nth term

a14 = 11 + (14 - 1)1 Substitute.

= 11 + 13

= 24 Simplify.

There are 24 seats in the 14th row.

B How many seats in total are in the first 14 rows?Find S14 using the formula for finding the sum of the first n terms.

S n = n(

a1 + an_2 ) Formula for first n terms

S 14 = 14(

11 + 24_2 ) Substitute.

= 14( 35_2 )

= 245 Simplify.

There are 245 seats in rows 1 through 14.

6. What if...? Suppose that each row after the first had 2 additional seats.

a. How many seats would be in the 14th row?

b. How many total seats would there be in the first 14 rows?

THINK AND DISCUSS 1. Compare an arithmetic sequence with a linear function.

2. Describe the effect that a negative common difference has on an arithmetic sequence.

3. Explain how to find the 6th term in a sequence when you know the 3rd and 4th terms.

4. Explain how to find the common difference when you know the 7th and 12th terms of an arithmetic sequence.

5. GET ORGANIZED Copy and complete the graphic organizer. Write in each rectangle to summarize your understanding of arithmetic sequences.

37 seats

336 total seats

Lesson 12-3 883





and INTERVENTIONSummarizeAsk students how to identify an arithmetic sequence. Then have them explain how to use the general rule for arithmetic sequences to find the nth term. An arithmetic sequence has a common first difference; to find the nth term, use a n = a 1 + (n - 1) d, where a 1 is the first term and d is the common difference.

Close3

Students sometimes confuse the sign of the common difference in an arithmetic sequence. Encourage stu-dents to check that the rule actually produces the terms of a sequence.

Technology Before Example 5, you may wish to review how to use the

sum(seq( features on a graphing calculator or have students look back at the Technology Lab that precedes this lesson. Calculators can help stu-dents check their work on difficult problems.

Answers to Think and DiscussPossible answers:

1. An arithmetic sequence is a linear func-tion with the domain restricted to the natural numbers.

2. decreases with each new term

3. Find d by subtracting a 4 - a 3 , use the general rule to solve for a 1 , and then evaluate for n = 6.

4. Substitute a 7 for a 1 and a 12 for a n in the general rule and solve for d.

5. See p. A13.


GUIDED PRACTICE 1. Vocabulary The expression 10 + 20 + 30 + 40 + 50 is an

−−−

? . (arithmetic sequenceor arithmetic series)


Determine whether each sequence could be arithmetic. If so, find the common difference and the next term.

2. 46, 39, 32, 25, 18, … 3. 28, 21, 15, 10, 6, … 4. 12_3

, 10_3

, 8_3

, 6_3

, 4_3

, …



5. 3, 8, 13, 18, … 6. 10, 9 3_4

, 9 1_2

, 9 1_4

, … 7. -3.2, -3.4, -3.6, -3.8, …


Find the missing terms in each arithmetic sequence.

8. 13, , , 25 9. 9, , , , 37 10. 1.4, , , , -1



11. a4 = 27 and a5 = 19 12. a3 = 12.2 and a4 = 12.6 13. a3 = -5 and a6 = -11

14. a10 = 100 and a20 = 50 15. a7 = -42 and a11 = -28 16. a4 = 3_4

and a8 = 1_2



17. S15 for 5 + 9 + 13 + 17 + � 18. ∑

k = 1

12

(-2 + 6k)

19. S18 for 3.2 + 2.9 + 2.6 + 2.3 + �


20. Salary Juan has taken a job with an initial salary of $26,000 and annual raises of $1250.

a. What will his salary be in his 6th year?

b. How much money in total will Juan have earned after six years?



21–23 1 24–26 2 27–29 3 30–32 4 33–35 5 36 6

Independent Practice Determine whether each sequence could be arithmetic. If so, find the common difference and the next term.

21. 288, 144, 72, 36, 18, … 22. -2, -12, -22, -32, -42, … 23. 0.99, 0.9, 0.81, 0.72, …


24. 12, 11.9, 11.8, 11.7, … 25. 2_5

, 3_5

, 4_5

, 1, … 26. -3.0, -2.5, -2.0, -1.5, …


27. 77, , , , 33 28. -29, , , -2 29. 2.3, , , , 1.5


30. a4 = 18.4 and a5 = 16.2 31. a4 = -2 and a8 = 46 32. a22 = -49 and a25 = -58


33. S15 for -18 + (-16) + (-14) + � 34. ∑

k = 1

20(88 - 3k) 35. ∑

k = 1

14

(14 -

1_2

k)

KEYWORD: MB7 12-3

KEYWORD: MB7 Parent

12-3 ExercisesExercises


Extra Practice

California Standards12.0, 22.0, 23.0

arithmetic series

arithmetic; -7; 11 not arithmetic arithmetic; - 2_3

; 2_3

38 8 1_4 -4.6

17, 21 16, 23, 30 0.8, 0.2, -0.4

-13-1714.6

105 -35

7_16

495 444

11.7

$32,250

$174,750

not arithmetic arithmetic; -10; -52 arithmetic; -0.09; 0.63

1112_5 2

66, 55, 44 -20, -11 2.1, 1.9, 1.7

0.8 94 -19

-60 1130143.5

884 Chapter 12

ExercisesExercises


Assignment Guide


If you finished Examples 1–3 Basic 21–28 Proficient 21–29 Advanced 21–29, 50

If you finished Examples 1–6 Basic 21–37, 39–51 odd,

52–57, 62–69 Proficient 21–47, 50–58, 62–69 Advanced 21–36 even, 37–69


12-3

One of London’s most recognizable landmarks, Big Ben sits atop the Palace of Westminister, where the British Parliament meets. The name Big Ben actually refers to the 13.8-ton bell that chimes the hours.

Clocks


36. Consumer Economics Clarissa is buying a prom dress on layaway. She agrees to make a $15 payment and increase the payment by $5 each week.

a. What will her payment be in the 9th week?

b. How much money in total will Clarissa have paid after 9 weeks?

37. Clocks A clock chimes every hour. The clock chimes once at 1 o’clock, twice at 2 o’clock, and so on.

a. How many times will the clock chime from 1 P.M. through midnight? in exactly one 24-hour period?

b. What if...? Another clock also chimes once on every half hour. How does this affect the sequence and the total number of chimes per day?


38. ∑

k = 1

16(555 - 11k) 39. ∑

k = 1

15(4 - 0.5k) 40. ∑

k = 1

18

(-33 + 5_

2k

)

41. S16 for 7.5 + 7 + 6.5 + 6.0 + � 42. S18 for 2 + 9 + 16 + 23 + �

43. Architecture The Louvre pyramid in Paris, France, is built of glass panes. There are 4 panes in the top row, and each additional row has 4 more panes than the previous row.

a. Write a series in summation notation to describe the total number of glass panes in n rows of the pyramid.

b. If the pyramid were made of 18 complete rows, how many panes would it have?

c. The actual pyramid has 11 panes less than a complete 18-row pyramid because of the space for the entrance. Find the total number of panes in the Louvre pyramid.

44. Physics Water towers are tall to provide enough water pressure to supply all of the houses and businesses in the area of the tower. Each foot of height provides 0.43 psi (pounds per square inch) of pressure.

a. Write a sequence for the pressure in psi for each foot of height.

b. What is the minimum height that supplies 50 psi, a typical minimum supply pressure?

c. What is the minimum height that supplies 100 psi, which is a typical maximum pressure?

d. Graph the sequence, and discuss the relationship between the height found for a pressure of 50 psi and the height found at 100 psi.

45. Exercise Sheila begins an exercise routine for 20 minutes each day. Each week she plans to add 5 minutes per day to the length of her routine.

a. For how many minutes will she exercise each day of the 6th week?

b. What happens to the length of Sheila’s exercise routine if she continues this increasing pattern for 2 years?

46. Geology Every year the continent of North America moves farther away from Europe.

a. How much farther from Europe will North America be in 50 years?

b. How many years until an extra mile is added? (Hint: 1 mi ≈ 1609 m)

$55$315

7384 0 -166.5

60 1107

0.43,0.86,1.29,1.72, ...

about 116.3 ft

about 232.6 ft

45 minutes45b. after 2 years, her exercise routine would be over 8 h long, which is not realistic

115 cm, or 1.15 m

≈ 69,956 yr

43a. ∑

k = 1

n

4k

673

684

Lesson 12-3 885

Students may not know how to approach Exercises 38–40. Emphasize that the formula for the sum of an arithmetic series requires three values: the value of the first term, the value of the last term, and the number of terms included. If they can find these three values, they can use the formula to find the sum.

Answers 37a. 78; 156

b. adds 1 to each term of the sequence; adds 24 to the total number per day

44d.

The height is twice as great because the graph represents a direct variation function.

12-3 PRACTICE C


48. Finance The starting salary for a summer camp counselor is $395 per week. In each of the subsequent weeks, the salary increases by $45 to encourage experienced counselors to work for the entire summer. If the salary is $710 in the last week of the camp, for how many weeks does the camp run?

49. Sports A town is planning a 5K race. The race route will begin at 1st street, travel 30 blocks down Main Street, and finish on 31st Street. The race planners want to have water stations at each turn. In addition, they want to place 5 more water stations evenly distributed between 1st Street and 31st Street on Main Street.

a. At what street intersections should the water stations be placed?

b. If each block is 0.1 mile, what is the maximum distance a runner will be from a water station while on Main Street?

50. Critical Thinking What is the least number of terms you need to write the general rule for an arithmetic sequence? How many points do you need to write an equation of a line? Are these answers related? Explain.

51. Write About It An arithmetic sequence has a positive common difference. What happens to the nth term as n becomes greater and greater? What happens if the sequence has a negative common difference?

52. Which sequence could be an arithmetic sequence?1_2

, 1_3

, 1_4

, 1_5

, … 2, 4, 8, 16, …

2.2, 4.4, 6.6, 8.8, … 2, 4, 7, 11, …

53. A catering company charges a setup fee of $45 plus $12 per person. Which of the following sequences accurately reflects this situation?

an = 45 + 12 (n - 1) an = 57 + 12n

45, 57, 69, 81, 93, … 57, 69, 81, 93, 105, …


You can make a simple tetrahedral kite with one peak by using 4 tetrahedrons. You can also make long kites with multiple peaks by successively adding 3 tetrahedrons as shown.

a. How many tetrahedrons are needed to make a kite with 20 peaks?

b. A kite maker wants to build one example of every kite with 1 to 20 peaks. How many tetrahedrons will be needed?

8 weeks

49a. 6th, 11th, 16th, 21st, and 26th Streets

0.25 mi

61

650

886 Chapter 12

Exercise 47 involves using an arithmetic series to identify the

number of tetrahedrons needed to make a kite. This exercise prepares students for the Concept Connection on page 888.

In Exercise 52, stu-dents who choose A, C, or D may not

recognize that arithmetic sequences must have a common difference. Remind students that they can check first differences to determine wheth-er any sequence could be arithmetic.

Answers 50. Possible answer: 2 terms;

2 points; yes, an arithmetic sequence is a linear function with a limited domain.

51. Possible answer: If d > 0, the value of the nth term increases toward infinity. If d < 0, the value of the nth term decreases toward negative infinity.

58a. a n = a 1 + (n - 1) d

−−−−−−−−−−−−−−−−

- ( a m = a 1 + (m - 1) d )

a n - a m = (n - 1) d - (m - 1) d

a n - a m = nd - d - md + d

a n - a m = nd - md

a n - a m = (n - m) d a n - a m

_ n - m = d

59. S n = n ( 2 a 1 + (n - 1) d

__ 2 ) ;

when you do not know the last term but know 2 of the 3 remaining variable quantities, S n , n, or a 1

61. The equality states that the value of a term, such as a 6 , is twice the value of the term that has half its term number, such as a 3 . This gives a 6 = 2 a 3 . For this to be true, the first term must equal the common difference, or a1 = d.

62.

63.

64.


a207se_c12l03_0879_0887.indd 886 3/15/07 2:09:51 PM


54. Which graph might represent the terms of an arithmetic sequence?

55. Given the arithmetic sequence 4, , , , 40, what are the three missing terms?

11, 22, 33 14, 24, 34

13, 22, 31 16, 24, 36

56. Which represents the sum of the arithmetic series 19 + 16 + 13 + 10 + 7 + 4?

∑

k = 1

6

19 - 3k ∑

k = 1

6(22 - 3k)

∑

k = 1

6

19 - 4k ∑

k = 1

6⎡

⎣ 22 -3(k - 1)⎤

⎦

57. Gridded Response What is the 13th term of the arithmetic sequence 54, 50, 46, 42, … .

CHALLENGE AND EXTEND 58. Consider the two terms of an arithmetic series an and am.

a. Show that the common difference is d = an - am_______n - m .

b. Use the new formula to find the common difference for the arithmetic sequence with a12 = 88 and a36 = 304.

59. Find a formula for the sum of an arithmetic sequence that does NOT include the last term. When might this formula be useful?

60. The sum of three consecutive terms of an arithmetic sequence is 60. If the product of these terms is 7500, what are the terms?

61. Critical Thinking What does a2n = 2 an mean and for what arithmetic sequences is it true?

SPIRAL REVIEWTell whether the function shows growth or decay. Then graph. (Lesson 7-1)

62. f (x) = 1.25 (0.75)

x63. f (x) = 1.43 (5.32)

x64. f (x) = 0.92 (0.64)

x

65. Sound The loudness of sound is given by L = 10 log (

I__I0)

, where L is the loudness of

sound in decibels, I is the intensity of sound, and I0 is the intensity of the softest audible sound. A sound meter at an auto race had a relative intensity of 1 0 9.2 I0 . Find the loudness of the sound in decibels. (Lesson 7-3)

Write each series in summation notation. (Lesson 12-2)

66. 1 + 1_2

+ 1_3

+ 1_4

+ 1_5

67. 4_5

+ 8_5

+ 12_5

+ 16_5

+ 4

68. -1 + 2 + 7 + 14 + 23 69. -

1_3

- 2_3

- 1 - 4_3

- 5_3

6

9

15, 20, 25 or 25, 20, 15

∑

k = 1

5 1_k ∑

k = 1

54k__5

∑

k = 1

5(k2 - 2) ∑

k = 1

5-

k_3

92 dB

62. decay

63. growth

64. decay

a207se_c12l03_0879_0887.indd 887 9/14/07 2:38:06 PM

Lesson 12-3 887


Math Background An alternative formula for the sum of an arithmetic

series is S n = n _ 2 [ 2 a 1 + (n - 1) d ].

Students are asked to find this for-mula in Exercise 59.

JournalHave students explain how to use the formula for the sum of the first n terms of an arithmetic series. Ask them to include an example.

Have students create an arithmetic sequence. Have them identify the common difference and several dif-ferent terms of the sequence. Have them show how the general rule and the sum formula work with the sequence they have created.

12-3

1. Determine whether the sequence could be arithmetic. If so, find the first difference and the next term.-1, -4, -7, -10, -13, …

yes; -3, -16

2. Find the 10th term of the arithmetic sequence-2, -5, -8, -11, -14, … . -29

3. Find the missing terms in the arithmetic sequence15, , , , 17.

15.5, 16, 16.5

4. Find the 6th term of the arith-metic sequence with a 9 = 64 and a 12 = 88. 40

5. Find the indicated sum for

∑

k = 1

12 (15 - 4k) . -132

6. The side section of an audito-rium has 12 seats in the first row and 3 additional seats in each subsequent row. How many seats are in the 10th row? How many seats in total are in the first 10 rows? 39; 255


a207te_c12_l3_879-887.indd 887a207te_c12_l3_879-887.indd 887 9/18/07 4:02:38 PM9/18/07 4:02:38 PM

2 N D P R I N T

Exploring Arithmetic Sequences and SeriesGo Fly a Kite! Alexander Graham Bell, the inventor of the telephone, is also known for his work with tetrahedral kites. In 1902, Bell used the kites to prove that it is possible to build an arbitrarily large structure that will fly. The kites are made up of tetrahedrons (four-sided triangular figures) with two sides covered with fabric. As shown in the figure, the size of a tetrahedral kite is determined by how many layers it has.

1. The first layer of a tetrahedral kite has 1 tetrahedron, the second layer has 3 tetrahedrons, and the third layer has 6 tetrahedrons. Write a sequence that shows how many tetrahedrons are in each of the first 10 layers.

2. Write a recursive formula for the sequence.

3. Write an explicit rule for the nth term of the sequence.

4. How many tetrahedrons are there in the 25th layer?

5. Write a series in summation notation that gives the total number of tetrahedrons in a kite with 25 layers.

6. Evaluate the series in problem 5 to find the total number of tetrahedrons in a kite with 25 layers. (Hint: Use the properties

∑

k = 1

n

c a k = c∑

k = 1

n

ak and ∑

k = 1

n

( a k + bk) = ∑

k = 1

n

ak + ∑

k = 1

n

bk.)

7. You see someone flying a large tetrahedral kite at a kite festival. You look up and estimate that the bottom layer of the kite contains between 100 and 110 tetrahedrons. How many layers does the kite have? How many tetrahedrons did it take to build the kite?

888

SECTION 12A

1, 3, 6, 10, 15, 21, 28, 36, 45, 55

a n = an-1 + n a n = n(n + 1)_

2325

∑

k = 1

25 k(k + 1)______2

2925

14; 560

OrganizerObjective: Assess students’ ability to apply concepts and skills in Lessons 12-1 through 12-3 in a real-world format.

Online Edition

Problem Text Reference

1–4 Lesson 12-1

5–6 Lesson 12-2

7 Lessons 12-1, 12-2

INTERVENTION

Scaffolding Questions 1. What pattern do you notice in the first

few terms of the sequence? The first dif-ferences are 2, 3, and 4.

2. How can you find the 5th term when you know the 4th term? Add 5 to the 4th term.

3. What do the differences tell you about the sequence? It is quadratic.

4. How do you find the 25th term? Evaluate for n = 25.

5. How would you write the first few terms and the last term of this series in expand-ed notation? 1 + 3 + 6 + … + 325

6–7. How can you apply summation formu-las to find the sum? Rewrite the series as the sum of a linear and a quadratic series.

ExtensionIs it possible to make a tetrahedral kite in which the bottom layer has exactly 360 tetrahedrons? no

S E C T I O N

12A

888 Chapter 12

Ready to Go On? 889

Quiz for Lessons 12-1 Through 12-3

12-1 Introduction to SequencesFind the first 5 terms of each sequence.

1. an = 2_3

n 2. an = 4 n-1

3. a1 = -1 and an = 2 an-1 - 12 4. an = n2- 2n


5. 8, 11, 14, 17, 20, . . . 6. -2, -8, -18, -32, -50, . . .

7. 1000, 200, 40, 8, 8_5

, . . . 8. 437, 393, 349, 305, 261, . . .

9. A car traveling at 55 mi/h passes a mile marker that reads mile 18. If the car maintains this speed for 4 hours, what mile marker should the car pass? Graph the sequence for n hours, and describe its pattern.

12-2 Series and Summation NotationExpand each series and evaluate.

10. ∑

k = 1

4(-14 - 2k) 11. ∑

k = 1

4

(

k_k + 2)

12. ∑

k = 1

5

(-1) k( k 2 - 2)


13. ∑

k = 1

51__2 14. ∑

k = 1

40

k2 15. ∑k = 1

15

k

16. The first row of a theater has 20 seats, and each of the following rows has 3 more seats than the preceding row. How many seats are in the first 12 rows?

12-3 Arithmetic Sequences and SeriesFind the 8th term of each arithmetic sequence.

17. 10.00, 10.11, 10.22, 10.33, . . . 18. -5, -13, -21, -29, . . .

19. a2 = 57.5 and a5 = 80 20. a10 = 141 and a13 = 186


21. -23, , , -89 22. 31, , , , 79

Find the indicated sum for each arithmetic series

23. S10 for 40 + 30 + 20 + 10 + . . . 24. ∑

k = 5

8

4k

25. ∑k = 1

11

(0.5k + 5.5) 26. S14 for –6 -1 + 4 + 9 + . . .

27. Suppose that you make a bank deposit of $1 the first week, $1.50 the second week, $2 the third week, and so on. How much will you contribute to the account on the last week of the year (52nd week)? What is the total amount that you have deposited in the bank after one year?

SECTION 12A

2_3 , 4_

3 , 2, 8_

3 , 10_

31, 4, 16, 64, 256

-1, -14, -40, -92, -196

a n = 1000 ( 1_5 )

n-1

1 + 2 - 7 + 14 - 23 = -13

5_2 22,140 120

10.77 -61102.5

-45, -67 43, 55, 67

-50

93.5 371

$26.50; $715

-1, 0, 3, 8, 15

a n = 437 - 44(n-1)

111

104

5. a n = 3n + 5 6. a n = -2n2

mile marker 238

10. -16 + (-18) + (-20) + (-22) = -76 11. 1_3 + 1_

2 + 3_

5 + 2_

3 = 21_

10

438

Ready to Go On? 889

OrganizerObjective: Assess students’ mastery of concepts and skills in Lessons 12-1 through 12-3.

ResourcesAssessment Resources

Section 12A Quiz

INTERVENTION

ResourcesReady to Go On? Intervention and Enrichment Worksheets

Ready to Go On? CD-ROM

Ready to Go On? Online

Answers 9. For graph, see p. A46.

NOINTERVENE

YESENRICH

READY TO GO ON? Intervention, Section 12AReady to Go On?

Intervention Worksheets CD-ROM Online

Lesson 12-1 Prep for 22.0 12-1 Intervention Activity 12-1Diagnose and

Prescribe OnlineLesson 12-2 Prep for 23.0 12-2 Intervention Activity 12-2

Lesson 12-3 22.0, 23.0 12-3 Intervention Activity 12-3

Diagnose and Prescribe

READY TO GO ON? Enrichment, Section 12A


S E C T I O N

12A

One-Minute Section PlannerLesson Lab Resources Materials

Lesson 12-4 Geometric Sequences and Series • Find terms of a geometric sequence, including geometric means.

• Find the sums of geometric series.




Technology Lab Activities12-4 Technology Lab


12-5 Algebra Lab Explore Infinite Geometric Series • Use a sequence of squares to explore an infinite geometric series.

□✔ SAT-10 □

NAEP □✔ ACT

□✔ SAT □✔ SAT Subject Tests

Algebra Lab Activities12-5 Lab Recording Sheet

Requiredgraph paper, graphing calculator

Lesson 12-5 Mathematical Induction and Infinite Geometric Series • Find sums of infinite geometric series.

• Use mathematical induction to prove statements.

□✔ SAT-10 □

NAEP □✔ ACT

□✔ SAT □✔ SAT Subject Tests

Optionaldominoes or books, graphing calculator

Extension Area Under a Curve • Approximate area under a curve by using rectangles.

□ SAT-10

□ NAEP

□ ACT

□ SAT

□ SAT Subject Tests


MK = Manipulatives Kit

SECTION

12B

890A Chapter 12

Exploring Geometric Sequences and Series

890B

GEOMETRIC SEQUENCESLesson 12-4In a geometric sequence, the ratio of successive terms is a constant called the common ratio r (r ≠ ±1). The nth term of a geometric sequence can be found by multiplying the first term by r (n - 1) times; equiva-lently, an = a1rn-1. This general rule is analogous to the equation of an exponential function, y = abx, and a geometric sequence may be viewed as an exponen-tial function whose domain is restricted to the natural numbers.

For a1 > 0, the value of r determines the nature of the geometric sequence.

• When r > 1, the sequence represents exponential growth. The terms of the sequence become arbitrarily large.

• When 0 < r < 1, the sequence represents exponential decay. The terms of the sequence become arbitrarily small. That is, they approach (but never equal) 0.

• When r < 0, the signs of the terms alternate between positive and negative.

Given two terms of a geometric sequence, am and an, the relationship an = amrn-m may be used to calcu-late the common ratio. Notice that this formula holds whether m < n or m > n, and that the formula reduces to the general rule for the nth term when m = 1.

GEOMETRIC SERIES 22.0, 23.0

Lessons 12-4, 12-5A geometric series is a series whose terms form a geo-metric sequence. The partial sum Sn of the first n terms of a geometric series is given by Sn = a1 + a1r + a1r2

+

… + a1rn-1. Multiplying each side of this equation by r gives rSn = a1r + a1r2

+ … + a1r n-1+ a1r n.

Subtracting rSn from Sn leads to the general formula for the partial sum of the first n terms of a geometric series,

Sn = a1(

1 - rn_1 - r ).

It is also possible to find the sum S of an infinite geometric series when |r| < 1. In this case, the sum is defined to be the limit of the partial sums as napproaches infinity. That is, S = lim n → ∞

Sn.

When |r| < 1, as n becomes larger and larger, the val-ues of rn approach 0. Therefore, when n is very large, rn has a negligible effect on the value of Sn. Loosely speaking, rn “disappears” from the numerator as n

approaches infinity, leaving S =

a1____1 - r .

When |r| ≥ 1, the partial sums Sn increase without bound and the sum of the infinite series is not defined. Students will see a more rigorous treatment of these ideas when they study limits in a calculus course.

MATHEMATICAL INDUCTION 21.0

Lessons 12-4, 12-5Mathematical induction is a method of proof used to demonstrate that a statement is true for all natural numbers. The first known use of the technique is cred-ited to an Italian mathematician, Francesco Maurolico, who used mathematical induction in 1575 to prove that the sum of the first n odd numbers is n2. Students should be aware that although the name of the method includes the word induction, the method is an example of deductive reasoning rather than inductive reasoning.

To use mathematical induction to prove that a state-ment is true for all natural numbers n, two things must be shown.

1. The statement is true for n = 1.2. If the statement is true for any natural number k,

then the statement is true for k + 1.

The rationale for these two parts of the proof is analo-gous to the way in which a row of dominos may be toppled. The first step of a proof by mathematical induction corresponds to tipping over the first domino. The second step of the proof says that when one dom-ino falls, it always knocks over the next one. Putting these two ideas together shows that the entire chain of dominos falls. In other words, the statement that is true for n = 1 is in fact true for all subsequent values of n.

Math Background: Teaching the Standards

Who uses this?Sporting-event planners can use geometric sequences and series to determine the number of matches that must be played in a tournament. (See Example 6.)


Serena Williams was the winner out of 128 players who began the 2003 Wimbledon Ladies’ Singles Championship. After each match, the winner continues to the next round and the loser is eliminated from the tournament. This means that after each round only half of the players remain.

The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence , the ratio of successive terms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is 1__

2.

Term a1 a2 a3 a4

Value 128 64 32 16

Ratios 64_128

= 1_2

32_64

= 1_2

16_32

= 1_2

Recall that exponential functions have a common ratio. When you graph the ordered pairs (n, an) of a geometric sequence, the points lie on an exponential curve as shown. Thus, you can think of a geometric sequence as an exponential function with sequential natural numbers as the domain.

1E X A M P L E Identifying Geometric Sequences

Determine whether each sequence could be geometric or arithmetic. If possible, find the common ratio or difference.


1a. 1_4

, 1_12

, 1_36

, 1_108

, . . .

1b. 1.7, 1.3, 0.9, 0.5, . . .

1c. -50, -32, -18, -8, . . .

12-4 Geometric Sequences and Series

ObjectivesFind terms of a geometric sequence, including geometric means.

Find the sums of geometric series.

Vocabularygeometric sequencegeometric meangeometric series

A 8, 12, 18, 27, . . .

8, 12, 18, 27

Diff. 4 6 9

Ratio 3_2

3_2

3_2

It could be geometric, with r = 3__

2.

B 8, 16, 24, 32, . . .

8, 16 24, 32

Diff. 8 8 8

Ratio 2 3_2

4_3

It could be arithmetic, with d = 8.

C 6, 10, 15, 21, . . .

6, 10, 15, 21

Diff. 4 5 6

Ratio 5_3

3_2

7_5

It is neither.


23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.Also covered: 22.0

geometric; r = 1_3

arithmetic; d = -0.4neither

890 Chapter 12

Introduce1

12-4 OrganizerObjectives: Find terms of a geometric sequence, including geometric means.

Find the sums of geometric series.

Technology LabIn Technology Lab Activities

Online EditionTutorial Videos, TechKeys

Warm UpSimplify.

1. 18 _ 27

2 _ 3 2. 128 (

3 _ 4 ) 96

Evaluate.

3. (-2) 8 256 4. ( 1 _ 2 )

6 1 _

64

Solve for x.

5. x 2 = 4 _ 25

x = ± 2 _ 5


A math student came home at 3 A.M. His upset parents exclaimed, “You’re late! You said you’d be home by 11:45!” The math student calmly replied, “I’m right on time. I said I’d be home by a quarter of twelve.”

MotivateAsk students to consider the following situation: a collectible purchased for $100 increases in value by either $15 per year or 10% per year. Have students discuss which increase they would pre-fer and why. Possible answer: The 10% increase would be better after 10 years or more. Explain that this situation can compare an arithmetic sequence to a geometric sequence.


California Standards 23.0

Also covered: 22.0Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

12- 4 Geometric Sequences and Series 891

Each term in a geometric sequence is the product of the previous term and the common ratio, giving the recursive rule for a geometric sequence.

n th term Common ratio

First term

You can also use an explicit rule to find the nth term of a geometric sequence. Each term is the product of the first term and a power of the common ratio as shown in the table.

Tennis Players in Each Round of Wimbledon

Round 1 2 3 4 n

Players 128 64 32 16 an

Formula a1 = 128 (

1_2)

0a2 = 128

(

1_2)

1a3 = 128

(

1_2)

2a4 = 128

(

1_2)

3an = 128

(

1_2)

n-1

This pattern can be generalized into a rule for all geometric sequences.

The nth term an of a geometric sequence is

an = a1 r n-1,

where a1 is the first term and r is the common ratio.

General Rule for Geometric Sequences

2E X A M P L E Finding the n th Term Given a Geometric Sequence

Find the 9th term of the geometric sequence -5, 10, -20, 40, -80, . . . .

Step 1 Find the common ratio.

r =

a2_a1

= 10_-5

= -2

Step 2 Write a rule, and evaluate for n = 9.

an = a1 rn-1 General rule

a 9 = -5(-2)

9-1 Substitute -5 for a1, 9 for n, and -2 for r.

= -5(256) = -1280

The 9th term is -1280.

Check Extend the sequence.

a5 = -80 Given

a6 = -80(-2) = 160

a7 = 160 (-2) = -320

a8 = -320(-2) = 640

a9 = 640 (-2) = -1280 ✔

Find the 9th term of each geometric sequence.

2a. 3_4

, -

3_8

, 3_16

, -

3_32

, 3_64

, . . . 2b. 0.001, 0.01, 0.1, 1, 10, . . .

3_1024

100,000

Lesson 12-4 891

Example 1


A. 100, 93, 86, 79, …

arithmetic; d = -7

B. 180, 90, 60, 15, … neither

C. 5, 1, 0.2, 0.04, …

geometric; r = 0.2

Example 2

Find the 7th term of the geomet-ric sequence 3, 12, 48, 192, … .

12,288


Additional Examples


EXAMPLE 1

• How would you determine wheth-er a sequence is arithmetic or geometric?

• How could you try to find a pos-sible rule for a sequence that is neither arithmetic nor geometric?

EXAMPLE 2

• How could you check the value you found for r, the common ratio?

• How could you predict the sign of the 9th term before you calculate it?

Technology To display the terms of a geometric sequence on a graph-

ing calculator, enter the first term,

press , press , enter

the common ratio, and press repeatedly.

Guided InstructionReview with students the domain and range for sequences with values in the form (n, a n ). Help students recall that the domain, or set of values for n, is the natu-ral numbers, while the range is some set of real numbers.

Inclusion Make sure that students realize that r cannot be 0 and that the value of a n

cannot be 0.

Teach2

Through Graphic Organizers

Have students make a table for geometric sequences, labeling the rows so that the first row shows the values of n, the second row shows the value of a n , and the third row shows the ratio between terms. The table for Example 2 is shown below.

n 1 2 3 …

a n -5 10 -20 …

r -2 -2 … …


3E X A M P L E Finding the nth Term Given Two Terms

Find the 10th term of the geometric sequence with a5 = 96 and a7 = 384.


a 7 = a5 r (7-5) Use the given terms.

a 7 = a5 r2 Simplify.

384 = 96r2 Substitute 384 for a7 and 96 for a5.

4 = r2 Divide both sides by 96.

±2 = r Take the square root of both sides.

Step 2 Find a1 .

Consider both the positive and negative values for r.

a n = a1 rn-1 an = a1rn-1 General rule

96 = a1(2)

5-1 or 96 = a1(-2)

5-1 Use a5 = 96 and r = ±2.

6 = a1 6 = a1

Step 3 Write the rule and evaluate for a10 .

Consider both the positive and negative values for r.

a n = a1r n-1 an = a1rn-1 General rule

a n = 6(2)

n-1 or an = 6(-2)

n-1 Substitute for a1 and r.

a 10 = 6 (2)

10-1 a10 = 6 (-2)

10-1 Evaluate for n = 6.

a 10 = 3072 a10 = -3072

The 10th term is 3072 or -3072.

Find the 7th term of the geometric sequence with the given terms.

3a. a4 = -8 and a5 = -40 3b. a2 = 768 and a4 = 48

Geometric means are the terms between any two nonconsecutive terms of a geometric sequence.

If a and b are positive terms of a geometric sequence with exactly one term between them, the geometric mean is given by the following expression.

√ �� ab

Geometric Mean

4E X A M P L E Finding Geometric Means

Find the geometric mean of 1_2

and 1_32

.

√ �� ab = √

��

( 1_2 ) ( 1_

32 )

= √

��

1_64

=

1_8

Use the formula.

4. Find the geometric mean of 16 and 25.

When given two terms of a sequence, be sure to consider positive and negative values for r when necessary.

-1000 3_4 or - 3_

4

20

892 Chapter 12

Example 3

Find the 8th term of the geomet-ric sequence with a 3 = 36 and a 5 = 324. 8748 or -8748

Example 4

Find the geometric mean of 4 _ 9

and 25 _ 36

. 5 _ 9


Additional Examples


EXAMPLE 3

• Will each term in a geometric sequence always be greater than the first term? Explain.

• Is the first term in a geometric sequence always positive? Explain.

EXAMPLE 4

• If a and b are nonconsecutive terms of a geometric sequence and there is exactly one term between them, can ab be less than 0? Explain.


The indicated sum of the terms of a geometric sequence is called a geometric series . You can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn as shown.

S n = a1 + a1r + a1r 2+ � + a1rn-1

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

- rSn = - a1r - a1r 2- � - a1rn-1

- a1rn

S n - r S n = a1 - a1rn

S n (1 - r) = a1 (1- rn)

S n = a1( 1- rn_

1-r )

The partial sum Sn of the first n terms of a geometric series a1 + a2 + � + an

is given by

Sn = a1(

1 - rn_1 - r ), r ≠ 1

where a1 is the first term and r is the common ratio.

Sum of the First Terms of a Geometric Series

5E X A M P L E Finding the Sum of a Geometric Series

Find the indicated sum for each geometric series.

A S7 for 3 - 6 + 12 - 24 +. . . B ∑

k = 1

5

(

1_3)

k-1

Step 1 Find the first term.

a1 = (

1_3)

1-1=

(

1_3)

0= 1

Step 2 Find S5 .

Sn = a1 (

1 - rn_1 - r ) Sum formula

S5 = 1 (

1 - (

1__3 )

5

_

1 - (

1__3 )

)Substitute.

=

(

1 - (

1___243)_

2__3

)

= 242_243

· 3_2

= 121_81

≈ 1.49


✔


r =

a2_a1

=

-6_3

= -2

Step 2 Find S7 with a1 = 3, r = -2, and n = 7.

Sn = a1(

1 - rn_1 - r )

Sumformula

S7 = 3 (

1 - (-2)

7_1 - (-2)

)Substitute.

= 3 (1 - (-128)__

3 )

= 129


✔


5a. S6 for 2 + 1 + 1_2

+

1_4

+. . . 5b. ∑

k = 1

6

-3(2)

k-1

These sums are partial sums because you are finding the sum of a finite number of terms. In Lesson 12-5, you will find the sums of infinite geometric series.

63_16 -189

Lesson 12-4 893

When applying the sum formula for geometric series, students may subtract the values in the numera-tor before applying the exponent. Remind them that exponents are evaluated before addition and sub-traction in the order of operations.

Example 5


A. S 8 for 1 + 2 + 4 + 8 + 16 + … 255

B. ∑

k = 1

6 (

1 _ 2 )

k - 1 63 _

32 ≈ 1.97


Additional Examples


EXAMPLE 5

• What are the first 5 terms of the geometric series in Example 5B? Explain.

• Can the sum of the first 5 terms of a geometric series ever be equal to zero? Explain.


6E X A M P L E Sports Application

The Wimbledon Ladies’ Singles Championship begins with 128 players. The players compete until there is 1 winner. How many matches must be scheduled in order to complete the tournament?

Step 1 Write a sequence.

Let n = the number of rounds,

a n = the number of matches played in the nth round, and

Sn = the total number of matches played through n rounds.

an = 64 (

1_2 )

n-1 The first round requires 64 matches, so a1 = 64. Each successive match requires 1__

2 as many, so r = 1__

2 .

Step 2 Find the number of rounds required.

1 = 64 (

1_2)

n-1The final round will have 1 match, so

substitute 1 for an.

1_64

= (

1_2)

n-1Isolate the exponential expression by dividing by 64.

( 1_2 )

6

= (

1_2)

n-1Express 1__

64 as a power of 1__

2: 1__

64= (

1__2)

6.

6 = n - 1 Equate the exponents.

7 = n Solve for n.

Step 3 Find the total number of matches after 7 rounds.

S7 = 64(

1 - ( 1_2 )

7

_

1 - ( 1_2 ) )

= 127 Sum function for geometric series.

127 matches must be scheduled to complete the tournament.

6. Real Estate A 6-year lease states that the annual rent for an office space is $84,000 the first year and will increase by 8% each additional year of the lease. What will the total rent expense be for the 6-year lease?

THINK AND DISCUSS 1. Find the next three terms of the geometric sequence that begins

3, 6, . . . . Then find the next three terms of the arithmetic sequence that begins 3, 6, . . . .

2. Compare the geometric mean of 4 and 16 with the mean, or average.

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, summarize your understanding of geometric sequences.

For a review of solving exponential equations, see Lesson 7-5.

$616,218.04

894 Chapter 12

Example 6

An online video game tourna-ment begins with 1024 players. Four players play in each game, and in each game, only the win-ner advances to the next round. How many games must be played to determine the winner? 341


Additional Examples


EXAMPLE 6

• How can you tell how many games or matches will be required for the first round?

• How do you solve an exponential equation? Explain the steps.

Inclusion The notation of sequences and series can be confusing for some stu-

dents. Encourage students to write down the value of each variable as they read through the problem. Then they can analyze the given information and determine what formulas to use.



Monitor During the LessonCheck It Out! Exercises, SE pp. 890–894

Questioning Strategies, TE pp. 891–894


and INTERVENTIONSummarizeAsk students to explain the difference between arithmetic and geometric sequences. Ask students to explain the difference between a geometric sequence and a geometric series. Arithmetic sequences have a common difference, while geometric sequences have a common ratio. A geometric series is the indicated sum of the terms of such a sequence.


1. geometric: 12, 24, 48; arithmetic: 9, 12, 15

2. Mean: 10; geometric mean: 8; either of the terms can be found by adding or subtracting 6 from the mean; either of the terms can be found by multiplying or dividing the geometric mean by 2.

3. See p. A13.



ExercisesExercises

GUIDED PRACTICE 1. Vocabulary The term between two given terms in a geometric sequence is the

−−−

? . (geometric mean or geometric series)



2. -10, -12, -14, -16, … 3. 1_2

, 1, 2, 3, … 4. -320, -80, -20, -5, …



5. 2, 6, 18, 54, 162, … 6. 5000, 500, 50, 5, 0.5, … 7. -0.125, 0.25, -0.5, 1, -2, …



8. a4 = -12, a5 = -4 9. a2 = 4, a5 = 108 10. a3 = 3, a5 = 12


Find the geometric mean of each pair of numbers.

11. 6 and 3_8

12. 2 and 32 13. 12 and 192



14. S6 for 2 + 0.2 + 0.02 + � 15. ∑

k = 1

5

(-3)

k-1

16. S 5 for 12 - 24 + 48 - 96 + � 17. ∑

k = 1

9

256(

1_2)

k-1


18. Salary In his first year, a math teacher earned $32,000. Each successive year, he earned a 5% raise. How much did he earn in his 20th year? What were his total earnings over the 20-year period?



19–21 1 22–25 2 26–28 3 29–31 4 32–35 5 36 6

Independent Practice Determine whether each sequence could be geometric or arithmetic. If possible, find the common ratio or difference.

19. -36, -49, -64, -81, … 20. -2, -6, -18, -54, … 21. 2, 7, 12, 17, …


22. 1_2

, 1_10

, 1_50

, 1_250

, 1_1250

, … 23. 3, -6, 12, -24, 48, …

24. 3200, 1600, 800, 400, 200, … 25. 8, 24, 72, 216, 648, …


26. a4 = 54, a5 = 162 27. a5 = 13.5, a6 = 20.25 28. a4 = -4, a6 = -100


29. 9 and 1_9

30. 18 and 2 31. 1_5

and 45


32. S6 for 1 + 5 + 25 + 125 + … 33. S8 for 10 + 1 + 1_10

+ 1_100

+ …

34. ∑

k = 1

6

-1(

1_3)

k-1 35. ∑

k = 1

7

8(10) k-1

KEYWORD: MB7 12-4

KEYWORD: MB7 Parent

12-4


Extra Practice


geometric mean

arithmetic; d = -2 neither geometric, r = 1_4

39,366 640.000005

- 4_3 324 24 or -24

3_2 8 48

2.22222

132

61

$80,862.41; $1,058,110.53

neither geometric; r = 3 arithmetic; d = 5

1_781,250

12.5

768

1458 30.375 500 or -500

1 6 3

511

52,488

3906

- 364_243

8,888,888

11.111111

ExercisesExercises

Assignment Guide


If you finished Examples 1–3 Basic 19–28 Proficient 19–28, 51 Advanced 19–28, 50–51


68–75 Proficient 19–39, 43–51, 57–65,

68–75 Advanced 20–36 even, 40–75


12-4

Lesson 12-4 895


36. Genealogy You have 2 biological parents, 4 biological grandparents, and 8 biological great grandparents.

a. How many direct ancestors do you have in the 6 generations before you? 12 generations?

b. What if...? How does the explicit rule change if you are considered the first generation?

Given each geometric sequence, (a) write an explicit rule for the sequence, (b) find the 10th term, and (c) find the sum of the first 10 terms.

37. 1_16

, 1_8

, 1_4

, 1_2

, … 38. 4, 0.4, 0.04, 0.004, … 39. 8, 16, 32, 64, …

40. -22, -11, -

11_2

, -

11_4

, … 41. 162, -54, 18, -6, … 42. 12.5, 62.5, 312.5, 1562.5, …

43. Collectibles Louis bought a vintage Rolling Stones concert shirt for $20. He estimates that the shirt will increase in value by 15% per year.

a. How much is the shirt worth after 4 years? after 8 years?

b. Does the shirt increase more in value during the first 4 years or the second 4 years? Explain.

44. College Tuition New grandparents decide to pay for their granddaughter’s college education. They give the girl a penny on her first birthday and double the gift on each subsequent birthday. How much money will the girl receive when she is 18? 21? Will the money pay for her college education? Explain.

45. Technology You receive an e-mail asking you to forward it to 5 other people to ensure good luck. Assume that no one breaks the chain and that there are no duplications among the recipients. How many e-mails will have been sent after 10 generations, including yours, have received and sent the e-mail?

46. Fractals The Sierpinski carpet is a fractal based on a square. In each iteration, the center of each shaded square is removed.

a. Given that the area of the original square is 1 square unit, write a sequence for the area of the nth iteration of the Sierpinski carpet.

b. In which iteration will the area be less than 1__2 of the original area?

47. Paper A piece of paper is 0.1 mm thick. When folded, the paper is twice as thick.

a. Studies have shown that you can fold a piece of paper a maximum of 7 times.How thick will the paper be if it is folded on top of itself 7 times?

b. Assume that you could fold the paper as many times as you want. How many folds would be required for the paper to be taller than Mount Everest (8850 m)?

48. Measurement Several common U.S. paper sizes are shown in the table.

a. Examine the length and width measures for the different paper sizes. What interrelationships do you observe?

b. How are the areas for each paper size (from A to E) mathematically related and what name is this relationship given?

Common U.S. Paper Sizes

U.S. Paper Size Dimensions (in.)

A (letter)

B (ledger)

C

D

E

81__2

× 11

11 × 17

17 × 22

22 × 34

34 × 44

The most expensive rock’n’roll memorabilia ever purchased was John Lennon’s Rolls Royce, with a price tag of $2.2 million.

Collectibles

126; 8190

The exponent changes from n to n - 1 such that an = 2 n becomes an = 2 n-1 .

$34.98; $61.18

2,441,406

27 folds

47a. 12.8 mm

an = ( 8_9 )

n-1

7th

896 Chapter 12

Diversity For Exercise 36, discuss the term biological parents, and be sure stu-

dents understand that this limitation on the number of parents is used for mathematical reasons. Many people have varying numbers of stepparents or foster parents, as well as other extended family, but every person has exactly two biological parents.

Answers 37. a n = 1 _

16 (2) n-1 ; a 10 = 32;

S 10 = 1023 _ 16

≈ 63.94

38. a n = 4 ( 1 _ 10

) n-1

;

a 10 = 0.000000004;

S 10 = 4.444444444

39. a n = 8 (2) n-1 ; a 10 = 4096;

S 10 = 8184

40. a n = -22 ( 1 _ 2 )

n-1

; a 10 = - 11 _ 256

;

S 10 = - 11,253

_ 256

≈ -43.96

41. a n = 162 (- 1 _ 3 )

n-1 ; a 10 = - 2 _

243 ;

S 10 = 121 121 _ 243

≈ 121.5

42. a n = 12.5 (5) n-1 ;

a 10 = 24,414,062.5;

S 10 = 30,517,575

43b. The shirt’s value increases more in the second 4 years because the increase is a percentage of the value, which becomes greater every year.

44. $1310.72; $10,485.76; the money will probably not pay for 4 years of college.


12-4 PRACTICE C

12-4 PRACTICE B

12-4 PRACTICE A



50. Biology The population growth of bacteria in a petri dish each hour creates a geometric sequence. After 1 hour there were 4 bacteria cells, and after 5 hours there were 324 cells. How many cells were found at hours 2, 3, and 4?

51. Critical Thinking Find an arithmetic sequence, a geometric sequence, and a sequence that is neither arithmetic nor geometric that begins 1, 4, ….

52. Finance Suppose that you pay $750 in rent each month. Suppose also that your rent is increased by 10% each year thereafter.

a. Write a series that describes the total rent paid each year over the first 5 years, and find its sum.

b. Use sigma notation to represent the series for the total rent paid each year over the first 10 years, and evaluate it.

53. Music The frequencies produced by playing C notes in ascending octaves make up a geometric sequence. C0 is the lowest C note audible to the human ear.

a. The note commonly called middle C is C4. Find the frequency of middle C.

b. Write a geometric sequence for the frequency of C notes in hertz where n = 1 represents C1.

c. Humans cannot hear sounds with frequencies greater than 20,000 Hz. What is the first C note that humans cannot hear?

54. Medicine During a flu outbreak, a hospital recorded 16 cases the first week, 56 cases the second week, and 196 cases the third week.

a. Write a geometric sequence to model the flu outbreak.

b. If the hospital did nothing to stop the outbreak, in which week would the total number infected exceed 10,000?

55. Graphing Calculator Use the SEQ and SUM features to find each indicated sum of the geometric series 8 + 6 + 4.5 + � to the nearest thousandth.

a. S10 b. S20 c. S30 d. S40

e. Does the series appear to be approaching any particular value? Explain.

56. Critical Thinking If a geometric sequence has r > 1, what happens to the terms as n increases? What happens if 0 < r < 1?

57. Write About It What happens to the terms of a geometric sequence when the first term is tripled? What happens to the sum of this geometric sequence?

58. Find the sum of the first 6 terms for the geometric series 4.5 + 9 + 18 + 36 + �.

67.5 144 283.5 445.5

Scale of C’s

Note Frequency (Hz)

C0 16.24

C1 32.7

C2 65.4

C3 130.8

C4


A movie earned $60 million in its first week of release and $9.6 million in the third week of release. The sales each week can be modeled by a geometric sequence.

a. Estimate the movie’s sales in its second week of release.

b. By what percent did the sales decrease each week?

c. In what week would you expect sales to be less than $1 million?

d. Estimate the movie’s total sales during its 8-week release period.

$24 million60%

week 6

about $99.93 million

12, 36, 108

30.198 31.899 31.994 32.00055e. Yes, the series appears to be approaching 32.

about 261.6 Hz

a n = 16 (3.5) n-1

C11

a n ≈ 16.3 (2)n

week 6


Exercise 49 involves applying the model of a geometric

sequence and finding a partial sum of a geometric series. This exercise prepares students for the Concept Connection on page 908.

In Exercise 58, it may be helpful to write the 5th and 6th

terms. Both A and B can be elimi-nated because the 6th term is 144 and the sum must be greater than this. Estimation shows that C is the correct choice.

For students who have difficulty with Exercise 59, remind them that the graph of a geometric sequence is similar in shape to the graph of an exponential function. Only choice G is close to this shape.

Answers 48a. Possible answer: The lon-

ger dimension becomes the shorter dimension in each subsequent paper size, and the shorter dimension is doubled to become the longer dimension.

b. Possible answer: Each paper size has twice as much area as the previous size. The areas form a geometric sequence with a common ratio of 2.

51. Possible answer: arithmetic: 1, 4, 7, 10, 13…; geometric: 1, 4, 16, 64, 256, …; neither: 1, 4, 9, 16, 25, …

52a. 9000 + 9900 + 10,890 + 11,979 + 13,176.90; $54,945.90

b. ∑

k = 1

10 9000 (1.1) k-1 ; $143,436.82

56. Possible answer: For r > 1, the sequence increases or decreases without bound as n increases. For 0 < r < 1, the sequence decreases toward a specific value as n increases.

57. Possible answer: Each term in the geometric sequence is tripled, and the sum of the series is also tripled.

Lesson 12-4 897


59. Which graph might represent the terms of a geometric sequence?

60. Find the first 3 terms of the geometric sequence with a7 = -192 and a9 = -768.

3, -6, 12 -3, 6, -12 or -3, -6, -12

-3, 12, -48 3, -12, 48 or -3, -12, -48

61. Which represents the sum of the series 10 - 15 + 22.5 - 33.75 + 50.625?

∑

k = 1

5

10(

3_2)

k-1 ∑

k = 1

5

-10(

3_2)

k-1

∑

k = 1

5

10(-

3_2)

k-1 ∑

k = 1

5

10(-

3_2)

k

62. Short Response Why does the general rule for a geometric sequence use n - 1 instead of n? Explain.

CHALLENGE AND EXTENDGraphing Calculator For each geometric sequence, find the first term with a value greater than 1,000,000.

63. a1 = 10 and r = 2 64. a1 = 1_4

and r = 4 65. a1 = 0.01 and r = 3.2

66. The sum of three consecutive terms of a geometric sequence is 73.5. If the product of these terms is 2744, what are the terms?

67. Consider the geometric sequence whose first term is 55 with the common ratio 1 + √ � 5______2 .

a. Find the next 5 terms rounded to the nearest integer.

b. Add each pair of successive terms together. What do you notice?

c. Make a conjecture about this sequence.

SPIRAL REVIEWIdentify the zeros and asymptotes of each function. (Lesson 8-2)

68. f (x) = x2

+ 2x - 3__x + 1

69. f (x) = x + 5_x2

- x - 6 70. f (x) = x

2- 16_4x

71. Shopping During a summer sale, a store gives a 20% discount on all merchandise. On Mondays, the store takes another 10% off of the sale price. (Lesson 9-4)

a. Write a composite function to represent the cost on Monday of an item with an original price of x dollars.

b. Find the cost on Monday of an item originally priced at $275.

Find the 10th term of each arithmetic sequence. (Lesson 12-3)

72. 78, 65, 52, 39, 26, … 73. 1.7, 7.3, 12.9, 18.5, 24.1, …

74. 9.42, 9.23, 9.04, 8.85, 8.66, … 75. 16.4, 26.2, 36, 45.8, 55.6, …

3.5, 14, 56, or 56, 14, 3.5

89, 144, 233, 377, 61067b. Their sum is the next term.

Beginning with 55, this sequence appears to round to the Fibonacci numbers.

f (x) = 0.9 (0.8x) = 0.72x$198

-39 52.17.71 104.6

898 Chapter 12

JournalHave students identify any confusion or difficulty they may have encoun-tered in working with geometric sequences and series. Have students describe strategies for correcting or preventing them.

Have students write the first 5 terms of a geometric sequence, give its rule in both recursive and explicit form, find the 10th term, and find the sum of the first 8 terms.

12-4

1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or differ-ence. 1 _

3 , 2, 12, 72, …

geometric; r = 6

2. Find the 8th term of the geometric sequence 1, -2, 4, -8, … . -128

3. Find the 9th term of the geometric sequence with a 2 = 0.3 and a 6 = 0.00003.

0.00000003

4. Find the geometric mean of 1 _ 2 and 18. 3

5. Find the indicated sum for

the geometric series ∑

k = 1

4

3 k - 1 .

40

6. A math tournament begins with 81 students. Students compete in groups of 3, with 1 person from each trio going on to the next round until there is 1 winner. How many matches must be played in order to complete the tournament? 40


Answers 62. Possible answer: The common ratio is

not used to find the first term, so it is used only n - 1 times to find the nth term.

63. a 18 = 1,310,720

64. a 12 = 1,048,576

65. a 17 ≈ 1,208,925.82

68. zeros: 1 and -3; vertical asymptote: x = -1

69. zero: -5; vertical asymptotes: x = -2 and x = 3; horizontal asymptote: y = 0

70. zeros: 4 and -4; vertical asymptote: x = 0

Explore Infinite Geometric SeriesYou can explore infinite geometric series by using a sequence of squares.

Activity

1. On a piece of graph paper, draw a 16 × 16 unit square. Note that its perimeter is 64 units.

2. Starting at one corner of the original square, draw a new square with side lengths half as long, or in this case, 8 × 8 units. Note that its perimeter is 32 units.

3. Create a table as shown at right. Fill in the perimeters and the cumulative sum of the perimeters that you have found so far.

Try This

1. Copy the table, and complete the first 6 rows.

2. Use summation notation to write a geometric series for the perimeters.

3. Use a graphing calculator to find the sum of the first 20 terms of the series.

4. Make a Conjecture Make a conjecture about the sum of the perimeter series if it were to continue indefinitely.

5. Evaluate 64_1 -

1__2

. How does this relate to your answer to Problem 4?

6. Copy and complete the table by finding the area of each square and the cumulative sums.

7. Use summation notation to write a geometric series for the areas.

8. Use a graphing calculator to find the sum of the first 10 terms of the series.

9. Make a Conjecture Make a conjecture about the sum of the area series if it were to continue indefinitely.

10. Evaluate 256_1 -

1__4

. How does this relate to your answer to

Problem 9?

11. Draw a Conclusion Write a formula for the sum of an infinite geometric sequence.

Square Perimeter Sum

16 × 16 64 64

8 × 8 32 96

4 × 4 ■ ■

2 × 2 ■ ■

1 × 1 ■ ■

1_2

× 1_2 ■ ■

Square Area Sum

16 × 16 ■ ■

8 × 8 ■ ■

4 × 4 ■ ■

2 × 2 ■ ■

1 × 1 ■ ■

1_2

× 1_2 ■ ■

12-5


12- 5 Algebra Lab 899

California Standards23.0 Students derive the summation formulas for arithmetic

series and for both finite and infinite geometric series.

∑

k = 1

n

64( 1_2 )

k-1

about 127.99988

Possible answer: The sum of the perimeters is 128.

128; it is equal to the sum of the perimeters.

about 341.333

341 1_3 ; it is equal to the sum of the areas.

S = a1_

1 - r

256 256

64 320

16 336

4 340

1 341

1_4 341 1_

4

16 112

8 120

4 124

2 126

Organizer

Answers to Try This 7. ∑

k = 1

n

256 ( 1 _ 4 )

k-1

9. Possible answer: The sum of the areas

is 341 1 _ 3 .

12-5 Algebra Lab 899

Objective: Use a sequence of squares to explore infinite geometric series.

Materials: graph paper, graphing calculator

Online Edition

ResourcesAlgebra Lab Activities

12-5 Lab Recording Sheet

TeachDiscussHave students describe the pattern as the squares’ sizes decrease.

Alternative ApproachModel the set with squares on a tiled floor or with square blocks.

CloseKey ConceptWhen ⎪r⎥ < 1, a formula can be used to find the sum of an infinite geo-metric sequence.

AssessmentJournal Have students explain why ⎪r⎥ < 1 must be true to find the sum of an infinite geometric sequence.


California StandardsAlgebra 2 23.0


In Lesson 12-4, you found partial sums of geometric series. You can also find the sums of some infinite geometric series. An infinite geometric series has infinitely many terms. Consider the two infinite geometric series below.

S n = 1_2

+ 1_4

+ 1_8

+ 1_16

+ 1_32

+ � Rn = 1_32

+ 1_16

+ 1_8

+ 1_4

+ 1_2

+ �

Partial Sums

n 1 2 3 4 5 6

Sn1_2

3_4

7_8

15_16

31_32

63_64

Partial Sums

n 1 2 3 4 5 6

Rn1_

32 3_

32 7_

32 15_

32 31_

32 63_

32

Notice that the series Sn has a common ratio of 1__2 and the partial sums get closer

and closer to 1 as n increases. When ⎪r⎥ < 1 and the partial sum approaches a fixed number, the series is said to converge . The number that the partial sums approach, as n increases, is called a limit .

For the series Rn, the opposite applies. Its common ratio is 2, and its partial sums increase toward infinity. When ⎪r⎥ ≥ 1 and the partial sum does not approach a fixed number, the series is said to diverge .

1E X A M P L E Finding Convergent or Divergent Series

Determine whether each geometric series converges or diverges.

A 20 + 24 + 28.8 + 34.56 + � B 1 + 1_3

+ 1_9

+ 1_27

+ 1_81

+ �

r = 24_20

= 1.2, ⎪r⎥ ≥ 1 r = 1__3_1

= 1_3

, ⎪r⎥ < 1

The series diverges and does The series converges and not have a sum. has a sum.


1a. 2_3

+ 1 + 3_2

+ 9_4

+ 27_8

+ � 1b. 32 + 16 + 8 + 4 + 2 + �

12-5 Mathematical Induction and Infinite Geometric Series

ObjectivesFind sums of infinite geometric series.

Use mathematical induction to prove statements.

Vocabularyinfinite geometric seriesconvergelimitdivergemathematical induction

Why learn this?You can use infinite geometric series to explore repeating patterns. (See Exercise 58.)


21.0 Students apply the method of mathematical induction to prove general statements about the positive integers.Also covered: 22.0, 23.0

divergesconverges

900 Chapter 12

Introduce1

12-5 OrganizerObjectives: Find sums of infinite geometric series.

Use mathematical induction to prove statements.

Online EditionTutorial Videos, Interactivity, TechKeys

Warm UpEvaluate.

1. 3 __ 5 _

6 1 _

10 2. 7 _

1 - 0.3 10

3. Write 0.6 as a fraction in simplest form. 3 _

5

4. Find the indicated sum for the geometric series

∑

k = 1

7

3 (-1) k - 1 . 3


Cat Theorem: A cat has nine tails.

Proof: No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.

MotivateSet up a row of about 5 dominoes (or books set on edge), and discuss what must happen to make the last one fall if you knock over only the first one. Help students see that there are two conditions involved–the first domino must fall, and each dom-ino must be placed so that when the one before it falls, it too will fall. Explain that this type of reason-ing is used in proof by induction.


California StandardsAlgebra 2 21.0Also covered: 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

12-5 Mathematical Induction and Infinite Geometric Series 901

If an infinite series converges, we can find the sum. Consider the series Sn = 1__

2 + 1__

4 + 1__8 + 1__

16 + 1__

32 + � from the previous page. Use the formula for the

partial sum of a geometric series with a1 = 1__2 and r = 1__

2 .

Sn = a1 (

1 - r n_1 - r ) = 1_

2 (

1 - (

1_2 )

n

_

1 -

1_2

)=

1(1 -

(

1_2 )

n

)__

2(

1_2)

= 1 -

(

1_2 )

n

_1

= 1 - (

1_2)

n

Graph the simplified equation on a graphing calculator. Notice that the sum levels out and converges to 1.

As n approaches infinity, the term (1__2)

n approaches zero. Therefore, the sum of

the series is 1. This concept can be generalized for all convergent geometric series and proved by using calculus.

The sum of an infinite geometric series S with common ratio r and ⎪r⎥ < 1 is

S =

a1_1 - r

,

where a1 is the first term.

Sum of an Infinite Geometric Series

2E X A M P L E Finding the Sums of Infinite Geometric Series

Find the sum of each infinite geometric series, if it exists.

A 5 + 4 + 3.2 + 2.56 + �

r = 0.8 Converges: ⎪r⎥ < 1

S = a1_

1 - rSum formula

= 5_1 - 0.8

= 5_0.2

= 25

Check Graph y = 5 (1 - (0.8)

x_

1 - 0.8 ) on a graphing

calculator. The graph approaches y = 25. ✔

B ∑

k = 1

∞

2_ 3 k-1

∑

k = 1

∞

2_ 3 k-1

= 2_1

+ 2_3

+ 2_9

+ � Evaluate.

r = 2__3_2

= 2_6

= 1_3

Converges: ⎪r⎥ < 1

S = a1_

1 - r= 2_

1 -

1__3

= 2_2__3

= 6_2

= 3

Check Graph y = 2 (

1 - (

1__3)

x

_1 -

1__3

) on a graphing

calculator. The graph approaches y = 3. ✔


2a. 25 - 5 + 1 -

1_5

+ 1_25

- … 2b. ∑

k = 1

∞

(

2_5)

k

You can graph a geometric series by using the sum formula from Lesson 12-4:

Sn = a1(

1 - rn_____1 - r ) with

Sn as y and n as xand with values substituted for rand a1.

2_3

2a. 125_6

Lesson 12-5 901

Students may attempt to find the sum of an infinite series before knowing that the series converges. Remind stu-dents that only convergent geometric series have a finite sum.

Example 1

Determine whether each geometric series, converges or diverges.

A. 10 + 1 + 0.1 + 0.01 + … converges

B. 4 + 12 + 36 + 108 + … diverges

Example 2


A. 1 - 0.2 + 0.04 - 0.008 + … 0.8 − 3

B. ∑

k = 1

∞

( 1 _ 2 )

k - 1 2


Additional Examples


EXAMPLE 1

• What do you look for in the graph of an infinite geometric series to help you determine whether it converges? Explain.

EXAMPLE 2

• How does a graph help you check the sum of an infinite geometric series?

Guided InstructionIntroduce the concepts of convergent and divergent series, and use several examples and graphs to reinforce the point. As stu-dents apply the sum formula, encourage them to check their answers by finding some partial sums. Explain that mathe-matical induction is an important method of proving that formulas work. Walk stu-dents through each step of the process because this will be new for most students.

Teach2

Advanced Learners/GATE

Have each student create or give each student an infinite geometric series that converges. Have them expand the series to 10 terms. Then have them create a table and a graph of the cumulative sums. Have them use their table and graph to estimate the sum of the series. Then have them use the formula to find the actual sum.


You can use infinite series to write a repeating decimal as a fraction.

3E X A M P L E Writing Repeating Decimals as Fractions

Write 0.232323. . . as a fraction in simplest form.

Step 1 Write the repeating decimal as an infinite geometric series.

0.232323… = 0.23 + 0.0023 + 0.000023 + � Use the pattern for the series.


r = 0.0023_0.23

= 1_100

, or 0.01 ⎪r⎥ < 1; the series converges to a sum.

Step 3 Find the sum.

S = a1_

1 - rApply the sum formula.

= 0.23_1 - 0.01

= 0.23_0.99

= 23_99

Check Use a calculator to divide the fraction 23_99

. ✔

3. Write 0.111… as a fraction in simplest form.

You have used series to find the sums of many sets of numbers, such as the first 100 natural numbers. The formulas that you used for such sums can be proved by using a type of mathematical proof called mathematical induction .

To prove that a statement is true for all natural numbers n,

Step 1 The base case: Show that the statement is true for n = 1.

Step 2 Assume that the statement is true for a natural number k.

Step 3 Prove that the statement is true for the natural number k + 1.

Proof by Mathematical Induction

4E X A M P L E Proving with Mathematical Induction

Use mathematical induction to prove that the sum of the first n natural

numbers is 1 + 2 + 3 + + n = n(n + 1)_______

2 .

Step 1 Base case: Show that the statement is true for n = 1.

1 = n(n + 1)_

2 =

1(1 + 1)_2

= 2_2

= 1 The base case is true.


1 + 2 + 3 + � + k = k(k + 1)_

2 Replace n with k.

Recall that every repeating decimal, such as 0.232323..., or 0.

−−

23, is a rational number and can be written as a fraction.

1_9

Math Background There are other methods for finding a repeating decimal.

For example, let x = 0. −−

23 .

100x = 23.232323…

−−−−−−−−−−−−−−

-x = -0.232323…

99x = 23

x = 23 _

99

902 Chapter 12

Technology On a graphing calculator, the , Frac command can be used to

convert some decimals to fractions. To convert repeating decimals, stu-dents must enter at least 11 digits.

Example 3

Write 0. −−

63 as a fraction in simplest form. 7 _

11

Example 4

Use mathematical induction to prove that 1 + 4 + 7 + … +

(3n - 2) = n (3n - 1)

_ 2 .

1. 1 = 1 (3 - 1)

_ 2 = 1.

2. 1 + 4 + … + (3k - 2) =

k (3k - 1)

_ 2 .

3. Add 3 (k + 1) - 2 to each side.

1 + … + 3k - 2 + 3 (k + 1) - 2

= k (3k - 1)

_ 2 + 3 (k + 1) - 2

= k (3k - 1)

_ 2 +

6 (k + 1) - 4 __

2

= 3 k 2 - k + 6k + 2

__ 2

= 3 k 2 + 5k + 2

__ 2

= (k + 1) (3k + 2)

__ 2

= (k + 1) (3k + 3 - 1)

__ 2

= (k + 1) ⎡

⎣ 3 (k + 1) - 1 ⎤

⎦

___ 2 .


Additional Examples


EXAMPLE 3

• How can you estimate the repeat-ing decimal for a fraction?

EXAMPLE 4

• If a statement is true for n = 1, n = 2, and n = 3, does that mean that it is true for every natural number n? Explain.


Step 3 Prove that it is true for the natural number k + 1.

1 + 2 + � + k = k(k + 1)_

2

1 + 2 + � + k + (k + 1) = k(k + 1)_

2 + (k + 1) Add the next term

(k + 1) to each side.

= k(k + 1)_

2 +

2(k + 1)_2

Find the common denominator.

= k(k + 1) + 2 (k + 1)__

2 Add numerators.

= (k + 1)(k + 2)__

2 Factor out k + 1.

= (k + 1)⎡

⎣ (k + 1) + 1⎤

⎦

__2

Write with k + 1.

Therefore, 1 + 2 + 3 + � + n = n(n + 1)_

2 .

4. Use mathematical induction to prove that the sum of the first nodd numbers is 1 + 3 + 5 + � + (2n - 1) = n2 .

Mathematical statements that seem to be true may in fact be false. By finding a counterexample, you can disprove a statement.

5E X A M P L E Using Counterexamples

Identify a counterexample to disprove 2 n ≥ n2 , where n is a real number.

2 0 ≥ (0)

2 2 1 ≥ (1)

2 2 4 ≥ (4)

2 2 -1≥ (-1)

2

1 ≥ 0 ✔ 2 ≥ 1 ✔ 24 ≥ 16 ✔ 1_2

≥ 1 ✘

2 n ≥ n2 is not true for n = -1, so it is not true for all real numbers.

5. Identify a counterexample to disprove a2_

2 ≤ 2a + 1,

where a is a real number.

THINK AND DISCUSS 1. Explain how to determine whether a geometric series converges or

diverges.

2. Explain how to represent the repeating decimal 0.8 −

3 as an infinite geometric series.

3. GET ORGANIZED Copy and complete the graphic organizer. Summarize the different infinite geometric series.

Often counter-examples can be found using special numbers like 1, 0, negative numbers, or fractions.

when a = 5: 5 2_2

?≤ 2 (5) + 1

12.5 ≤ 11

Lesson 12-5 903

Example 5

Identify a counterexample to disprove a 3 > a 2 , where a is a whole number. Possible answer: a = 0 or a = 1, so it is not true for all whole numbers


Additional Examples


EXAMPLE 5

• How many counterexamples does it take to disprove a statement? Explain.

Reading Math Remind students that an ellipsis (…) at the end of a series

indicates that it is an infinite series.

Answers to Check It Out 4. 1. 1 = 1 2

2. 1 + 3 + … + (2k -1) = k 2

3. 1 + 3 + … + (2k - 1)

+ [2 (k + 1) -1] = k 2 + [2 (k + 1) -1] = k 2 + 2k + 1

= (k + 1) 2





and INTERVENTIONSummarizeReview infinite geometric series, stressing that only convergent series have a finite sum. Have students list the steps of a proof by mathematical induction.


1. Find the common ratio. If ⎪r⎥ < 1, the series converges. If ⎪r⎥ ≥ 1, the series diverges.

2. Write 0.8 − 3 as 0.8 plus the infinite geo-metric series 0.03 + 0.003 + … .

3. See p. A13.


ExercisesExercises12-5KEYWORD: MB7 12-5

GUIDED PRACTICE 1. Vocabulary An infinite geometric series whose sum approaches

a fixed number is said to−−−

? . (converge or diverge)



2. 1 -

1_3

+ 1_9

-

1_27

+ 1_81

+ � 3. 1 - 5 + 25 - 125 + 625 + �

4. 27 + 18 + 12 + 8 + �



5. 3_4

+ 1_2

+ 1_3

+ 2_9

+ � 6. ∑

k = 1

∞

4(0.25)

k 7. 800 + 200 + 50 + �


Write each repeating decimal as a fraction in simplest form.

8. 0.888 … 9. 0. −−

56 10. 0.131313 …


11. Use mathematical induction to prove that the sum of the first n even numbers is 2 + 4 + 6 + � + 2n = n(n + 1) .


Identify a counterexample to disprove each statement, where n is a real number.

12. n4≥ 1 13. log n > 0 14. n3

≤ 3 n2



15–17 1 18–20 2 21–23 3 24 4 25–27 5

Independent Practice Determine whether each geometric series converges or diverges.

15. 3 + 3_5

+ 3_25

+ 3_125

+ 3_625

+ � 16. 5 + 10 + 20 + 40 + �

17. 2 - 4 + 8 - 16 + 32 + …


18. ∑

k = 1

∞

60(

1_10)

k 19. 8_

5 - 4_

5 + 2_

5 - 1_

5 + … 20. ∑

k = 1

∞

3. 5 k


21. 0. −

6 22. 0.90909 . . . 23. 0.541541541 . . .

24. Use mathematical induction to prove

1_1(2)

+ 1_2(3)

+ 1_3(4)

+ …+ 1_

n(n + 1)

= n_n + 1

.

Identify a counterexample to disprove each statement, where a is a real number.

25. a3≠ - a2 26. a4

> 0 27. 5 a2> 2 a

28. /////ERROR ANALYSIS///// Two possible sums for the series 1__5 + 2__

5 + 4__5 + � are

shown. Which is incorrect? Explain the error.

KEYWORD: MB7 Parent


Extra Practice


converge

converges

converges

diverges

9_4 16_

3 1066 2_3

8_9

56_99

13_99

n = 1_2

n = -1 n = 5

converges diverges

diverges

20_3

16_15

No sum exists.

2_3 10_

11 541_

999

a = 0a = 0 a = 0

A, the sum formula was incorrectly applied to a divergent series.

25–27. Possibleanswers are given.

904 Chapter 12

12–5 ExercisesExercises

Assignment Guide


If you finished Examples 1–2 Basic 15–20, 30–31 Proficient 15–20, 30–31, 46 Advanced 15–20, 30–31, 46


50–54, 61–65, 71–76 Proficient 15–38, 42–54, 59–65,

71–76 Advanced 16–38 even, 39–76


Answers11. 1. 2 (1) = 1 (1+1) = 2

2. 2 + 4 + … + 2 (k) = (k) (k+1)

3. 2 + 4 + … + 2k + 2 (k+1)

= k (k + 1) + 2 (k + 1)

= k 2 + k + 2k + 2

= k 2 + 3k + 2

= (k + 1) (k + 2)



29. Art Ojos de Dios are Mexican holiday decorations. They are made of yarn, which is wrapped around sticks in a repeated square pattern. Suppose that the side length of the outer square is 8 inches. The side length of each inner square is 90% of the previous square’s length. How much yarn will be required to complete the decoration? (Assume that the pattern is represented by an infinite geometric series.)


30. 215 - 86 + 34.4 - 13.76 + � 31. 500 + 400 + 320 + �

32. 8 - 10 + 12.5 - 15.625 + � 33. ∑

k = 1

∞

-5(

1_8)

k-1

34. ∑

k = 1

∞

2(

1_4)

k-135. ∑

k = 1

∞

(

5_3)

k-1

36. -25 - 30 - 36 - 43.2 + � 37. ∑

k = 1

n

200(0.6)

k-1

38. Geometry A circle of radius r has smaller circles drawn inside it as shown. Each smaller circle has half the radius of the previous circle.

a. Write an infinite geometric series in terms of r that expresses the circumferences of the circles, and find its sum.

b. Find the sum of the circumferences for the infinite set of circles if the first circle has a radius of 3 cm.


39. 0. −

4 40. 0. −

9 41. 0. −−−

123

42. 0. −−

18 43. 0. −

5 44. 0. −−−

054

45. Music Due to increasing online downloads, CD sales have declined in recent years. Starting in 2001, the number of CDs shipped each year can be modeled by a geometric sequence.

a. Estimate the number of CDs that will be shipped in 2010.

b. Estimate the total number of CDs shipped from 2001 through 2010.

c. Suppose that the geometric series continued indefinitely. Find the total number of CDs shipped from 2001.

Use mathematical induction to prove each statement.

46. 1 + 2 + 4 + � + 2 n-1= 2 n - 1

47. 1 + 4 + � + n2=

n(n + 1)(2n + 1)__6

48. 1 (2) + 2 (3) + 2 (4) + � + n(n + 1) = n(n + 1)(n + 2)__

3

49. 1_2

+ 1_4

+ 1_8

+ � + (

1_2)

n

= 1 - (

1_2)

n

Co

mp

act

dis

cs (

mill

ion

s)

Year

Annual Compact DiscShipments

1000

2001

600

700

800

900

5002002 2003

881.9

811.1

745.9

320 in., or 26 2_3 ft

30. 1075_7 ≈ 153.57 No sum exists.

2500

- 40_7

8_3 No sum exists.

12π ≈ 37.70 cm

4_9

1

41_333

about 415.0 millionabout 6.2 billion

about 11 billion

No sum exists.500

2_11

5_9

2_37

38a. ∑

k = 1

n

2π r ( 1_2 )

k; 4πr

Lesson 12-5 905

Some students might have difficulty finding counterexamples in Exercises 25–27. Remind them that a real number can be a negative number, a fraction, or an irrational number such as √�2 .

Geometry For Exercise 38,you may wish to remind students of the formula

for the circumference of a circle: C = πd.

Answers 24. 1. 1_

1 (1 + 1) =

1_1 + 1

=1_2

2. 1_1 (2)

+…+1_

k (k + 1) =

k_k + 1

3. 1_1 (2)

+ … +1_

k (k + 1)

+1__

(k + 1) (k + 2)

=k_

k + 1+

1__ (k + 1) (k + 2)

=k (k + 2) __

(k + 1) (k + 2)

+1__

(k + 1) (k + 2)

= k 2 + 2k + 1__

(k + 1) (k + 2)

= (k + 1) (k + 1) __ (k + 1) (k + 2)

= (k + 1) _ (k + 2)

46. 1. 2 1-1 = 2 1 - 1 = 1

2. 1 + … + 2 k-1 = 2 k - 1

3. 1 + … + 2 k-1 + 2 k+1-1

= 2 k - 1 + 2 k+1-1

= 2 k - 1 + 2 k

= 2(2 k) - 1 = 2 k+1- 1

47–49. See p. A47.11-5 PRACTICE C


51. Game Shows Imagine that you have just won the grand prize on a game show. You can choose between two payment options as shown. Which would you choose, and why?

Identify a counterexample to disprove each statement, where x is a real number.

52. x4_

x3 ≤ 2x 53. x4

- 1 ≥ 0 54. ln x5> ln x

55. 2 x2≤ 3 x3 56. 2 x2

- x ≥ 0 57. 12x - x2> 25

58. Geometry The midpoints of the sides of a 12-inch square are connected to form another concentric square as shown. Suppose that this process is continued without end to form a sequence of concentric squares.

a. Find the perimeter of the 2nd square.

b. Find the sum of the perimeters of the squares.

c. Find the sum of the areas of the squares.

d. Write the sum of the perimeters in summation notation for the general case of a square with side length s. Then write the sum of the areas for the general case.

e. Which series decreases faster, the sum of the perimeters or the sum of the areas? How do you know?

59. Critical Thinking Compare the partial-sum Sn with the sum S for an infinite geometric series when a1 > 0 and r = 4__

5 . Which is greater? What if a1 < 0?

60. Write About It Why might the notation for a partial sum Sn change for the sum S for an infinite geometric series?

61. Which infinite geometric series converges?

∑

k = 1

∞

(

5_4)

k∑

k = 1

∞

5(

1_4)

k∑

k = 1

∞

1_4

(5)

k∑

k = 1

∞

(

1_4)

k5k

62. What is the sum of the infinite geometric series 1 -

1_2

+

1_4

-

1_8

+

1_16

+ �?

2 2_3

1_2

1_3

63. An infinite geometric series has a sum of 180 and a common ratio of 2__3. What is the

first term of the series?

60 120 270 540

$100,000 a year for 20 years

$1 million the first year and half of the previous year’s amount for eternity


A movie earned $80 million in the first week that it was released. In each successive week, sales declined by about 40%.

a. Write a general rule for a geometric sequence that models the movie’s sales each week.

b. Estimate the movie’s total sales in the first 6 weeks.

c. If this pattern continued indefinitely, what would the movie’s total sales be?

a n = 80 (0.6) n-1 , where an represents millions of dollarsabout $190.67 million

$200 million

x = -1 x = 1_2

x = 0

x = -2 x = 0x = 1_4

24 √ � 2 ≈ 33.94 in.96 + 48 √ � 2 ≈ 163.88 in.

288 i n 2

The subscript n refers to a finite number of terms, but an infinite geometric series has an infinite number of terms.

52–57.Possible answers given.

906 Chapter 12

Exercise 50 involves using a geometric sequence and sums

for the corresponding series. This exercise prepares students for the Concept Connection on page 908.

Geometry For Exercise 58, you may wish to explain that concentric

means having a common center.

In Exercise 61, if stu-dents have trouble identifying r from the

formula, encourage them to find a few terms of each series. They should see that choices A, C, and D all have common ratios that are greater than 1.

In Exercise 64, the value of y, as x increases, continues to increase without limit in both F and G, so those choices could not represent a converging series. In J, the value of y decreases without limit, so the cor-rect choice is H.

Answers 51. Possible answer: option B;

option B gives the winner a total of $2 million after 20 yr. Option A is an infinite geometric series with a sum of $2 million, but in reality you would never collect the entire amount.

58d. perimeter: ∑

k = 1

n

4s ( √ � 2

_ 2 )

k-1

;

area: ∑

k = 1

n

s 2 ( 1 _ 2 )

k-1

e. The sum of the areas decreases faster because it has a smaller common ratio (multiplier) than

the perimeters, or 1 __ 2 < √ � 2 ___ 2 .

59. For a 1 > 0, S > S n , and both sums are positive. For a 1 < 0, S n > S, and both sums are negative.




64. Which graph represents a converging infinite geometric series?

65. Extended Response Use mathematical induction to prove 3 + 5 + � + (2n + 1) = n(n + 2). Show all of your work.

CHALLENGE AND EXTENDWrite each repeating decimal as a fraction in simplest form.

66. 0.1 −

6 67. 0.41 −

6 68. 0.52 −−

86

69. Critical Thinking Can an infinite arithmetic series approach a limit like an infinite geometric series? Explain why or why not.

70. Geometry Consider the construction that starts with a 12-inch square and contains concentric squares as indicated. Notice that a spiral is formed by the sequence of segments starting at a corner and moving inward as each midpoint is reached. A second similar spiral determines the area shown in blue.

a. Use the sum of a series to find the length of the spiralindicated in red.

b. Use the sum of a series to find the polygonal area indicated in blue.

c. Is your answer to the sum of the polygonal area in part b reasonable? Explain.

SPIRAL REVIEW 71. Football A kickoff specialist kicks 80% of his kickoffs into the end zone. What is

the probability that he kicks at least 4 out of 5 of his next kickoffs into the end zone? (Lesson 11-6)

72. Geometry Consider the pattern of figures shown. (Lesson 12-3)

a. Find the number of dots in each of the next 3 figures in the pattern.

b. Write a general rule for the sequence of the number of dots in the nth figure.

c. How many dots will be in the 22nd figure in the pattern?

Determine whether each sequence could be geometric or arithmetic. If possible, find the common ratio or difference. (Lesson 12-4)

73. 297, 99, 33, 11, … 74. 4_3

, 8_3

, 4, 16_3

, …

75. 25, 100, 250, 1000, … 76. 4, 4.8, 5.76, 6.912, …

1_6 5_

122617_4950

73.728%

9, 11, 13an = 3 + 2 (n - 1)

45 dots

geometric; r = 1_3 arithmetic; d = 4_

3

neither geometric; r = 6_5

Yes; the area of the spiral is 1__4 the area of the largest square.

12 + 6 √ � 2 ≈ 20.485 in.36 i n 2

Lesson 12-5 907


JournalAsk students to describe the process of proving a statement by math-ematical induction.

Have students create a convergent geometric series and find its sum. Have them also write a divergent geometric series. Ask them to describe how a proof by induction is similar to a row of dominoes that falls if you knock down the first one.

12-5

Solve each equation.

1. Determine whether the geometric series 150 + 30 + 6 + … converges or diverges, and find the sum if it exists. converges; 187.5

2. Write 0. −−−−

0044 as a fraction

in simplest form. 4 _ 909

3. Either prove by induction or provide a counterexample to disprove the following statement: 1 + 2 + 3 + 4 +

… + n = n 2 + 1

_ 2 .

counterexample: n = 2


Answers 65. 1. 2 (1) + 1 = 1 (1 + 2) = 3

2. 3 + … + (2k + 1) = k (k + 2)

3. 3 + … + (2k + 1) +

[2 (k + 1) + 1] = k (k + 2) + [2 (k + 1) + 1] = k 2 + 2k + (2k + 2 + 1)

= k 2 + 4k + 3

= (k + 1) (k + 3)

= (k + 1) (k + 1 + 2)

69. No; the partial sums will approach infinity if d > 0 and negative infinity if d < 0.

Exploring Geometric Sequences and SeriesSticky Business Big-budget movies often have their greatest sales in the first weekend, and then weekend sales decrease with each passing week. After a movie has been released for a few weeks, movie studios may try to predict the total sales that the movie will generate.

1. Find the ratios of the sequences of weekend sales for Spider-Manand Spider-Man 2.

2. Write the rule for a geometric sequence that could be used to estimate the sales for Spider-Man in a given weekend.

3. Use the sequence from Problem 2 to predict Spider-Man’s weekend sales for weeks 4 and 5.

4. Write and evaluate a series in summation notation to find Spider-Man’stotal weekend sales for the first 5 weekends of its release.

5. Suppose that the series from Problem 4 continued infinitely. Estimate the total weekend sales for Spider-Man. The actual total weekend sales for Spider-Man were about $311.1 million. How does this compare with your estimate?

6. Would a geometric sequence be a good model for the weekend sales of Spider-Man 2? Justify your answer.

Weekend Box Office Sales (million $)

Weekend Spider-Man Spider-Man 2

1 114.9 115.8

2 71.4 45.2

3 45.0 24.8

908 Chapter 12

SECTION 12B

about $28.1 million and $17.5 million

∑

k = 1

5114.9(0.625) k -1 ;

about $277.2 million

About $306.4 million; the estimate is very close.

OrganizerObjective: Assess students’ ability to apply concepts and skills in Lessons 12-4 through 12-5 in a real-world format.

Online Edition

Problem Text Reference

1–4 Lesson 12-4

5 Lesson 12-5

6 Lessons 12-4, 12-5

Answers 1. Spider-Man: ≈ 0.62, ≈ 0.63;

Spider-Man 2: ≈ 0.39, ≈ 0.55

2. a n = 114.9 (0.625) n -1 , where a n represents sales in millions of dollars

6. Possible answer: No; the ratios for Spider-Man 2 are not close to being constant, so it is very hard to develop an accurate geometric sequence.

INTERVENTION

Scaffolding Questions 1. How do you find common ratios? Divide

each term by the previous term.

2. What must you know to write the rule for a geometric sequence? the first term and the common ratio

3. How can you find the 4th and 5th terms? Evaluate the series for n = 4 and n = 5.

4. What is the formula for the partial sum of a geometric series?

S n = a 1 ( 1 - r n _ 1 - r

)

5. What is the formula for the sum of an infinite geometric series?

S n = a 1 _

1 - r

6. What identifies a geometric sequence? common ratio

ExtensionHave students find the box office data for a recent hit movie and determine whether a geometric sequence would be a good model for its weekend sales. Check students’ work.

S E C T I O N

12B

908 Chapter 12

Ready to Go On? 909

Quiz for Lessons 12-4 Through 12-5

12-4 Geometric Sequences and SeriesFind the 8th term of each geometric sequence.

1. 2_5

, 6_5

, 18_5

, 54_5

, . . . 2. -16, -40, -100, -250, . . .

3. -1, 11, -121, 1331, . . . 4. 2, 20, 200, 2000, . . .

Find the 10th term of each geometric sequence with the given terms.

5. a1 = 3.3 and a2 = 33 6. a4 = -1 and a6 = -4

7. a6 = 20.25 and a8 = 9 8. a3 = 57 and a5 = 513


9. 1_3

and 1_27

10. 4.5 and 450 11. 32 and 1_8


12. S6 for 8 -16 + 32 -64 + � 13. S5 for 1 + 2_3

+ 4_9

+ 8_27

+ �

14. ∑

k = 1

7

(8) k 15. ∑

k = 1

5

18(

1_6)

k-1

16. The cost for electricity is expected to rise at an annual rate of 8%. In its first year, a business spends $3000 for electricity.

a. How much will the business pay for electricity in the 6th year?

b. How much in total will be paid for electricity over the first 6 years?

12-5 Mathematical Induction and Infinite Geometric SeriesFind the sum of each infinite series, if it exists.

17. 25 + 20 + 16 + 12.8 + � 18. 15 - 18 + 21.6 - 25.92 + �

19. ∑

k = 1

∞

(-1) k(

2_3)

k 20. ∑

k = 1

∞

4(0.22) k

Use mathematical induction to prove 4 + 8 + 12 + � + 4n = 2n(n + 1) .

21. Step 1 22. Step 2 23. Step 3

24. A table-tennis ball is dropped from a height of 5 ft. The ball rebounds to 60% of its previous height after each bounce.

a. Write an infinite geometric series to represent the distance that the ball travels after it initially hits the ground. (Hint: The ball travels up and down on each bounce.)

b. What is the total distance that the ball travels after it initially hits the ground?

SECTION 12B

4374_5

-9765.625

19,487,171

5. 3,300,000,000

-644

1_9

45 2

-168

2,396,744 1555_72

≈ 21.60

211_81

≈ 2.60

$4407.98

125 No sum exists.44_39

, or 1. −−−−

128205-0.4

15 ft

∑

k = 1

∞

2(3)(0.6)k -1

$22,007.79

20,000,000

±124,659

Ready to Go On? 909

OrganizerObjective: Assess students’ mastery of concepts and skills in Lessons 12-4 through 12-5.


Section 12B Quiz

INTERVENTION

ResourcesReady to Go On? Intervention and Enrichment Worksheets

Ready to Go On? CD-ROM

Ready to Go On? Online

Answers 21. 4 = 2 (1) (1 + 1) = 4

22. 4 + … + 4k = 2k (k + 1)

23. 4 + … + 4k + 4 (k + 1)

= 2k (k + 1) + 4 (k + 1)

= (k + 1) (2k + 4)

= 2 (k + 1) (k + 2)

NOINTERVENE

YESENRICH

READY TO GO ON? Intervention, Section 12BReady to Go On?

Intervention Worksheets CD-ROM Online

Lesson 12-4 22.0, 23.0 12-4 Intervention Activity 12-4 Diagnose and Prescribe OnlineLesson 12-5 21.0, 22.0, 23.0 12-5 Intervention Activity 12-5

Diagnose and Prescribe

READY TO GO ON? Enrichment, Section 12B


S E C T I O N

12B


EXTENSIONEXTENSION

Finding the area under a curve is an important topic in higher mathematics, such as calculus. You can approximate the area under a curve by using a series of rectangles as shown in Example 1.

1E X A M P L E Finding Area Under a Curve

Estimate the area under the curve f (x) = - 1__

2 x2 + 2x + 3 1__

8 over

0 ≤ x ≤ 5.

Graph the function. Divide the area into 5 rectangles, each with a width of 1 unit.

Find the height of each rectangle by evaluating the function at the center of each rectangle, as shown in the table.

Approximate the area by finding the sum of the areas of the rectangles.

A ≈ 1(4) + 1(5) + 1(5) + 1(4) + 1(2) = 20

The estimate of 20 square units is very close to the actual area of 1919__

24 square units, which can be found by using calculus.

1. Estimate the area under the curve f (x) = - x2+ 5x + 5.75

over 0 ≤ x ≤ 6. Use 6 intervals.

You can formalize the procedure for finding the area under the curve of a function by using the sum of a series.

Number of rectangles

Area under the curve

Width of each rectangle

Middle x-value of each rectangle

Height of each rectangle

2E X A M P L E Finding Area Under a Curve by Using a Series

Use the sum of a series to estimate the area under the curve f (x) = - x2 + 50x over 0 ≤ x ≤ 50.

Step 1 Graph the function.

Step 2 Divide the area into 5 rectangles, each with a width of 10 units.

ObjectiveApproximate area under a curve by using rectangles.

Area Under a Curve

x f(x)

0.5 4

1.5 5

2.5 5

3.5 4

4.5 2

ak f(ak)

a1 = 5 225

a2 = 15 525

a3 = 25 625

a4 = 35 525

a5 = 45 225

Area is measured in square units.


Extension of 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

53 square units

CHAPTER

MotivateAsk students to identify some figures for which they could easily find the area. square, triangle, rectangle, etc. Ask them to consider finding the area of a figure bounded by a curve, such as the entrance to a highway tunnel. Explain that series can be used to approximate areas such as these.

Introduce1

Guided InstructionAs you review the procedure for estimating the area under a curve, remind students that it is only an estimate. There are several different ways to draw rectangles that rep-resent the area under the curve. Emphasize that calculus can be used to find the actual area.

Teach2

12 OrganizerObjective: Approximate area under a curve by using rectangles.

Online Edition

Using the ExtensionIn Chapter 12, students learn about geometric sequences and series. In this extension, students use the par-tial sum of a series to estimate the area under a curve.

Technology The table function on a graphing calculator can be used to

find the height of each rectangle. Set Tblstart to the center of the first rectangle and the ΔTbl to the width of each rectangle.

Precalculus The exact area under a curve can be found by finding the

integral of a function over a given interval. On a graphing calculator,

press and then select

7:∫f (x) dx.

910 Chapter 12

California StandardsExtension of 22.0

Chapter 12 Extension: Area Under a Curve 911

Estimate the area under each curve. Use 4 intervals.

1. f (x) = -

1_2

x2+ 12x + 2 over 0 ≤ x ≤ 24 2. f (x) = - x2

+ 8x + 4 over 0 ≤ x ≤ 8

3. f (x) = -

1_4

x2+ 5x over 0 ≤ x ≤ 16 4. f (x) = -0.1x2

+ 20x over 0 ≤ x ≤ 200

Use the sum of a series to estimate the area under each curve. Use 5 intervals.

5. f (x) = - x2+ 10x + 5 over 0 ≤ x ≤ 10 6. f (x) = - x2

+ 30x over 0 ≤ x ≤ 20

7. f (x) = -

1_10

x2+ 400 over 0 ≤ x ≤ 50 8. f (x) = -0.2x 2

+ 28x + 300 over 0 ≤ x ≤ 150

9. Physics The graph shows a car’s speed versus time as the car accelerates. This realistic curve can be approximated by v(t) = -0.1t2

+ 7.3t, where v is the velocity in feet per second and t is the time in seconds.

a. Estimate the area under the curve for 0 ≤ t ≤ 35.

b. What does the area under the curve represent? Explain. (Hint: Consider the units of your answer to part a.)

10. Energy Conservation Daily electricity use peaks in the early afternoon and can be approximated by a parabola. Suppose that the rate of electricity use in kilowatts (kW) is modeled by the function f (x) = -1.25x2

+ 30x + 700, where x represents the time in hours.

a. Write a sum to represent the area under the curve for a domain of 0 ≤ x ≤ 24.

b. Estimate the area under the curve.

11. Write About It Explain how the units of the values on the x-axis and the units of the values on the y-axis can be used to find the units that apply to the area under a curve.

12. Critical Thinking For a given function and domain, how would increasing the number of rectangles affect the approximation of the area under the curve?

EXTENSION

ExercisesExercises

Step 3 Find the value of the function at the center of each rectangle, as shown in the table.

Step 4 Write the sum that approximates the area.

A ≈ 10 ∑

k = 1

5

f(ak)

= 10 ⎡

⎣ f (a1) + f (a2) + f

(a3)

+ f (a4) + f (a5)⎤

⎦

= 10 ⎡

⎣ (225) + (525) + (625) + (525) + (225) ⎤

⎦

= 10(2125) = 21,250

The estimated area is 21,250 square units.

2. Use the sum of a series to estimate the area under the curve f (x) = - x2

+ 9x + 3 over 0 ≤ x ≤ 9. Use 3 intervals.

155.25 square units

1236 square units 120 square units

304 square units 137,500 square units

220 square units 3360 square units

15,875 square units8. 137,250 square units

19,720 square units

SummarizeHave students discuss how the accuracy of the estimate changes as the width of the rectangles decreases and the number of rectangles increases.

Close3 Answers9a. about 3056 square units

b. Distance in feet traveled in t s; to find the area of each rectangle, you multi-ply speed (ft/s) by time (s) , and the result is a distance (ft) .

10a. Possible answer: A = 4 ∑

k = 1

6 f ( a k )

= 4 (755 + 835 + 875 + 875 + 835 + 755)

11. Possible answer: Because you find area by multiplying the length and the width, you multiply the units to find the units that apply to the area under the curve.

12. Possible answer: Increasing the number of rectangles would make the answer more accurate.

Students may be confused by the units of the answer in Exercises 9 and 10. Explain that the graphs are only abstract representations of a function and that the area under the curve is not measured in common units of area, such as square feet.

Example 1

Estimate the area under the

curve f (x) =-

1 _ 2 x 2 + 2x + 3 1 _

8

over -1 ≤ x ≤ 2. Use 3 intervals.

≈ 11 square units

Example 2

Use the sum of a series to esti-mate the area under the curve

f (x) = - x 2 + 50x over 0 ≤ x ≤ 20.

≈ 7360 square units


Additional Examples


EXAMPLE 1

• How would your estimate of the area change if you made rectangles whose upper left corners were points on the curve?

• How does the number of rectan-gles affect your accuracy?

EXAMPLE 2

• Could you use the same function to find the area over an interval that does not start at zero, such as 20 ≤ x ≤ 50? Explain.

Technology The area under a curve can be esti-mated on a graphing cal-

culator by using the sum(seq( commands. For Example 2, let Y1 = - x 2 + 50x, and enter 10* sum(seq Y1, X, 5, 45, 10).

Extension 911

Study Guide: Review


arithmetic sequence . . . . . . . . 879

arithmetic series . . . . . . . . . . . . 882

converge . . . . . . . . . . . . . . . . . . . 900

diverge . . . . . . . . . . . . . . . . . . . . . 900

explicit formula . . . . . . . . . . . . 863

finite sequence . . . . . . . . . . . . . 862

geometric mean . . . . . . . . . . . . 892

geometric sequence . . . . . . . . 890

geometric series . . . . . . . . . . . . 893

infinite geometric series . . . . . 900

infinite sequence . . . . . . . . . . . 862

iteration . . . . . . . . . . . . . . . . . . . 864

limit . . . . . . . . . . . . . . . . . . . . . . . 900

mathematical induction . . . . . 902

partial sum . . . . . . . . . . . . . . . . . 870

recursive formula . . . . . . . . . . . 862

sequence . . . . . . . . . . . . . . . . . . . 862

series . . . . . . . . . . . . . . . . . . . . . . 870

summation notation . . . . . . . . 870

term of a sequence . . . . . . . . . . 862

Complete the sentences below with vocabulary words from the list above.

1. A(n) −−−−−−

? has a common difference, and a(n) −−−−−−

? has a common ratio.

2. A series that has no limit −−−−−−

? , whereas a series that approaches a limit −−−−−−

? .

3. A(n) −−−−−−

? defines the nth term. A(n) −−−−−−

? defines the next term by using one or more of the previous terms.

4. A(n) −−−−−−

? continues without end, and a(n) −−−−−−

? has a last term.

5. Each step in a repeated process is called a(n) −−−−−−

? .


6. an = n - 9 7. an = 1_2

n2

8. an = (-

3_2)

n -1

9. a1 = 55 and an = an -1 - 2

10. a1 = 200 and an = 1_5

an -1

11. a1 = -3 and an = -3an -1 + 1


12. -4, -8, -12, -16, -20, . . .

13. 5, 20, 80, 320, 1280, . . .

14. -24, -19, -14, -9, -4, . . .

15. 27, 18, 12, 8, 16_3

, . . .

16. Sports Suppose that a basketball is dropped from a height of 3 ft. If the ball rebounds to 70% of its height after each bounce, how high will the ball reach after the 4th bounce? the 9th bounce?

■ Find the first 5 terms of the sequence witha1 = -52; an = 0.5an -1 + 2.

Evaluate the rule using each term to find the next term.

n 1 2 3 4 5

an -52 -24 -10 -3 0.5

■ Write an explicit rule for the nth term of 100, 72, 44, 16, -12, . . . .

Examine the differences or ratios. Terms 100 72 44 16 -12

1st differences 28 28 28 28

The first differences are constant, so the sequence is linear.

The first term is 100, and each term is 28 less than the previous term.

The explicit rule is an = 100 - 28(n - 1) .

12-1 Introduction to Sequences (pp. 862–868)

EXERCISESE X A M P L E S

Prep for 22.0

912 Chapter 12

OrganizerObjective: Help students organize and review key concepts and skills presented in Chapter 12.

Online EditionMultilingual Glossary

Resources

Multilingual Glossary Online

KEYWORD: MB7 Glossary

Lesson Tutorial Videos CD-ROM

Answers 1. arithmetic sequence; geometric

sequence

2. diverges; converges

3. explicit formula; recursive formula

4. infinite sequence; finite sequence

5. iteration

6. -8, -7, -6, -5, -4

7. 1 _ 2 , 2, 9 _

2 , 8, 25 _

2

8. 1,- 3 _ 2 , 9 _

4 , - 27 _

8 , 81 _

16

9. 55, 53, 51, 49, 47

10. 200, 40, 8, 8 _ 5 , 8 _

25

11. -3, 10 ,-29, 88, -263

12. a n = -4n

13. a n = 5 (4) n-1

14. a n = 5n - 29

15. a n = 27 ( 2 _ 3 )

n-1

16. 0.72 ft, or 8.6 in.;

0.12 ft, or 1.5 in.

C H A P T E R

12

Study Guide: Review 913


17. ∑

k = 1

4

k2(-1) k 18. ∑

k = 1

5

(0.5k + 4)

19. ∑

k = 1

5

(-1) k+1(2k - 1) 20. ∑

k = 1

4

5k_k2


21. ∑

k = 1

8

-5 22. ∑

k = 1

10

k2 23. ∑

k = 1

12

k

24. Finance A household has a monthly mortgage payment of $1150. How much is paid by the household after 2 years? 15 years?

■ Expand ∑

k = 1

5(-1)n+1(11 - 2n), and evaluate.

∑

k = 1

5

(-1) n+1(11 - 2n) = (-1) 2(11 - 2)

+ (-1) 3(11 - 4) + (-1) 4(11 - 6)

+ (-1) 5(11 - 8) + (-1) 6(11 - 10)

= 9 - 7 + 5 - 3 + 1

= 5 Simplify.

■ Evaluate ∑

k = 1

8

k2 .

Use summation formula for a quadratic series.

∑

k = 1

8

k2=

n(n + 1)(2n + 1)__6

=

8(8 + 1)(2 · 8 + 1)__6

=

72(17)_6

= 204

12-2 Series and Summation Notation (pp. 870–877)



25. 23, 19, 15, 11, . . . 26. 1_5

, 3_5

, 1, 7_5

, 9_5

, . . .

27. -9.2, -8.4, -7.6, -6.8, . . .

28. a3 = 1.5 and a4 = 5

29. a6 = 47 and a8 = 21

30. a5 = -7 and a9 = 13


31. S18 for -1 - 5 - 9 - 13 + �

32. S12 for 1_3

+ 1_6

+ 0 -

1_6

+ �

33. ∑

k = 1

15

(-14 + 3k)

34. ∑

k = 1

15

(

3_2

k + 10)

35. Savings Kelly has $50 and receives $8 a week for allowance. He wants to save all of his money to buy a new mountain bicycle that costs $499. Write an arithmetic sequence to represent the situation. Then find whether Kelly will be able to buy the new bicycle after one year (52 weeks) .

■ Find the 12th term for the arithmetic sequence 85, 70, 55, 40, 25, . . ..

Find the common difference: d = 70 - 85 = -15.


a12 = 85 + (12 - 1)(-15) Substitute.

= -80 Simplify.

■ Find ∑

k = 1

11(-2 - 33k).

Find the 1st and 11th terms.

a1 = -2 -33(1) = -35

a11 = -2 - 33(11) = -365

Find S11 .

Sn = n(

a 1 + an_2 ) Sum formula

S11 = 11(

-35 - 365_2 )

Substitute.

= -2200

12-3 Arithmetic Sequences and Series (pp. 879–887)


Prep for 23.0

22.0, 23.0


Answers 17. -1 + 4 - 9 + 16 = 10

18. 4.5 + 5.0 + 5.5 + 6.0 + 6.5 = 27.5

19. 1 - 3 + 5 - 7 + 9 = 5

20. 5 + 10 _ 4 + 15 _

9 + 20 _

16 = 125 _

12 ,

or 10 5 _ 12

21. - 40

22. 385

23. 78

24. $27,600; $207,000

25. -17

26. 21 _ 5

27. -1.2

28. 29.5

29. -18

30. 23

31. - 630

32. - 7

33. 150

34. 330

35. 50, 58, 66, 74, … ; no, because he will have a total savings of only $458



36. 40, 4, 0.4, 0.04, 0.004, . . .

37. 1_18

, 1_6

, 1_2

, 3_2

, . . .

38. -16, -8, -4, -2, . . .

39. -6, 12, -24, 48, . . .


40. a3 = 24 and a4 = 96

41. a1 = 2_3

and a2 = -

4_3

42. a4 = -1 and a6 = -4

43. a3 = 4 and a6 = 500


44. 10 and 2.5 45. 1_2

and 8

46. √ � 3_96

and √ � 3_6

47. 5_12

and 125_108


48. S5 for 1 + 1_3

+ 1_9

+ 1_27

+ �

49. S6 for -

4_5

+ 8 - 80 + 800 + �

50. ∑

k = 1

8(4) k -1

51. ∑

k = 1

7

-2(5) k -1

52. ∑

k = 1

6

60(-

1_2)

k -1

53. ∑

k = 1

5

18(

1_2)

k -1

54. Depreciation A new photocopier costs $9000 and depreciates each year such that it retains only 65% of its preceding year’s value. What is the value of the photocopier after 5 years?

55. Rent A one-bedroom apartment rents for $650 a month. The rent is expected to increase by 6% per year.

a. What will be the annual rent expense on the apartment after 5 years?

b. What will be the total amount spent on rent if a person rents the apartment for the entire 5-year period?

■ Find the 8th term of the geometric sequence 6, 24, 96, 384, . . ..

Find the common ratio. r = 24_6

= 4

Write a rule, and evaluate for n = 8.

an = a1rn -1 General rule

a8 = 6 (4)8 -1 = 98,304

■ Find the 8th term of the geometric sequence with a4 = -1000 and a6 = -40.


a6 = a4r(6 -4) Use the given terms.

-40 = -1000r2 Substitute.

1_25

= r2 Simplify.

±

1_5

= r

Step 2 Find a1 using both possible values for r.

-1000 = a1( 1_5 )

4-1

or -1000 = a1(- 1_5 )

4-1

a 1 = -125,000 or a1 = 125,000

Step 3 Write the rule and evaluate for a8 by using both possible values for r.

a n = a1rn –1 an = a1rn -1

a n = -125,000 ( 1_5 )

n-1

or a n = 125,000 (-

1_5 )

n-1

a 8 = -125,000(

1_5)

8 –1 a 8 = 125,000

(-

1_5)

8 -1

a 8 = -1.6 a8 = -1.6

■ Find ∑

k = 1

7

-2(5)k -1 .

Find the common ratio. r = a2_a1

=

-6_3

= -2

Find S7

Sn = a1(

1 - rn_1 - r )

Sum formula

S7 = 3 (

1 - (-2)

7_1 - (-2)

)Substitute.

= 3 (1 - (-128)_

3 ) = 129

12-4 Geometric Sequences and Series (pp. 890–898)


22.0, 23.0

914 Chapter 12

Answers 36. 0.000004

37. 243 _ 2

38. - 1 _ 8

39. 768

40. 98, 304

41. 512 _ 3

42. ±32

43. 62,500

44. 5

45. 2

46. √ � 3 _ 24

47. 25 _ 36

48. 121 _ 81

49. 72,727.2

50. 21,845

51. - 39,062

52. 315 _ 8 = 39.375

53. 279 _ 8 = 34.875

54. $1044.26

55a. $9847.32

b. $43,969.32


Find the sum of each infinite series, if it exists.

56. -2700 + 900 - 300 + 100 + �

57. -1.2 - 0.12 - 0.012 - 0.0012 + �

58. -49 - 42 - 36 - 216_7

+ �

59. 4 + 4_5

+

4_25

+

4_125

+ �

60. ∑

k = 1

∞

9_ 3 k

61. ∑

k = 1

∞

-7(

3_5)

k

62. ∑

k = 1

∞

(-1)

k +1(

1_ 8 k )

63. ∑

k = 1

∞

(

4_3)

k

Use mathematical induction to prove each statement.

64. 2 + 4 + 8 + � + 2 n = 2 n +1- 2

65. 1 + 5 + 25 + � + 5 n-1=

5 n - 1_4

66. 1_3

+

1_15

+ � +

1_4n2

- 1=

n_2n + 1

67. Recreation A child on a swing is let go from a vertical height so that the distance that he travels in the first back-and-forth swing is exactly 9 feet.

a. If each swing decreases the distance by 85%, write an infinite geometric series that expresses the distance that the child travels in feet.

b. What is the total distance that the child in the swing travels before the swing stops?

Find the sum of each infinite series, if it exists.

■ -9261 + 441 - 21 + 1 + �

r =

441_-9261

= - 1_21


S =

a1_1 - r

Sum formula

=

-9261_

1 - (- 1___21 )

=

-9261_22__21

= -

194,481_22

, or -8840.0−−

45

■ ∑

k = 1

∞

-5(

7_10)

k -1

= -5 -

35_10

-

245_100

+ � Evaluate.

r =

-

35__10_

-5=

7_10


S =

a1_1 - r

=

-5_1 - 7___

10

=

-5_3__

10

= -

50_3

, or -16.−

6

■ Use mathematical induction to prove

2 + 5 + + (3n - 1) = n_2

(3n + 1) .

Step 1 Base case: Show that the statement is true for n = 1.

2 = n_2

(3n + 1) = 1_2

(3 · 1 + 1) = 2 True


2 + 5 + � + (3k - 1) = k_2

(3k + 1) Replace nwith k.

Step 3 Prove that it is true for the natural number k + 1.

2 + 5 + . . . + (3k - 1) + 3(k + 1) -1 Add to both sides.

= k_2

(3k + 1) + 3(k + 1) -1

= k(3k + 1)_

2 + ( 3k + 3 -1) Multiply.

= 3k2+ k_

2 +

2(3k + 2)_2

Simplify and rewrite with like denominators.

= 3k2+ 7k + 4__

2 Add.

= (k + 1)(3k + 4)__

2 Factor.

= (k + 1)_

2 (3(k + 1) + 1) Write with k + 1.

12-5 Mathematical Induction and Infinite Geometric Series (pp. 900–907)

EXERCISESE X A M P L E S 21.0, 22.0, 23.0


Answers 56. -2025

57. - 4 _ 3 or -1. − 3

58. -343

59. 5

60. 4.5

61. - 21 _ 2

62. 1 _ 9

63. No sum exists.

64. 1. 2 1 = 2 1+1 - 2 = 2

2. 2 + 4 + … + 2 k = 2 k+1 - 2

3. 2 + 4 + … + 2 k + 2 k+1 = 2 k+1 - 2 + 2 k+1 = 2 ( 2 k+1 ) - 2 = 2 k+2 - 2

= 2 k+1+1 - 2

65. 1. 5 1-1 = 5 1 - 1 _ 4 = 1

2. 1 + … + 5 k-1 = 5 k - 1 _ 4

3. 1 + … + 5 k-1 + 5 k

= 5 k - 1 _ 4 + 5 k

= 5 k - 1 _ 4 +

4 ( 5 k ) _

4

= 5 k + 4 ( 5 k ) - 1

__ 4

= 5 ( 5 k ) - 1

_

4 = 5 k+1 - 1 _

4

66. 1. 1 _ 4 ( 1 2 ) - 1

= 1 _ 2 (1) + 1

= 1 _ 3

2. 1 _ 3 + … + 1 _

4k 2 - 1 = k _

2k + 1

3. 1 _ 3 + … + 1 _

4k 2 - 1

+ 1 __ 4 (k + 1) 2 - 1

= k _ 2k + 1

+ 1 __ 4 (k + 1) 2 - 1

= k _ 2k + 1

+ 1 __ 4 k 2 + 8k + 3

= k _ 2k + 1

+ 1 __ (2k + 1) (2k + 3)

= k (2k + 3)

__ (2k + 1) (2k + 3)

+ 1 __ (2k + 1) (2k + 3)

= k (2k + 3) + 1

__ (2k + 1) (2k + 3)

= 2 k 2 + 3k + 1

__ (2k + 1) (2k + 3)

= (2k + 1) (k + 1)

__ (2k + 1) (2k + 3)

= k + 1 __

2k + 2 + 1

= k + 1 __

2 (k + 1) + 1

67a. ∑

k = 1

∞

9 (0.85) k - 1

b. 60 ft



1. an = n2- 4 2. a1 = 48 and an = 1_

2 an–1 - 8


3. -4, -2, 0, 2, 4, . . . 4. 54, 18, 6, 2, 2_3

, . . .


5. ∑

k = 1

4

5 k3 6. ∑

k = 1

7

(-1) k+1(k)


7. -19, -13, -7, -1, . . . 8. a2 = 11.6 and a5 = 5

9. Find 2 missing terms in the arithmetic sequence 125, , , 65.


10. S20 for 4 + 7 + 10 + 13 + . . . 11. ∑

k = 1

12

(-9k + 8)

12. The front row of a theater has 16 seats and each subsequent row has 2 more seats than the row that precedes it. How many seats are in the 12th row? How many seats in total are in the first 12 rows?


13. 3_256

, 3_64

, 3_16

, 3_4

, . . . 14. a4 = 2 and a5 = 8

15. Find the geometric mean of 4 and 25.


16. S6 for 2 + 1 + 1_2

+

1_4

+ . . . 17. ∑

k = 1

6

250(-

1_5)

k-1

18. You invest $1000 each year in an account that pays 5% annual interest. How much is the first $1000 you invested worth after 10 full years of interest payments? How much in total do you have in your account after 10 full years?


19. 200 - 100 + 50 - 25 + . . . 20. ∑

k = 1

∞

2(

7_8)

k

Use mathematical induction to prove 1_2

+3_2

+5_2

+ +2n - 1_

2=

n2_2

.

21. Step 1

22. Step 2

23. Step 3

-3,0, 5, 12, 21 48, 16, 0, -8, -12

a n = 2n - 6 a n = 54 (

1_3 )

n-1

5 + 40 + 135 + 320 = 500 1 - 2 + 3 - 4 + 5 - 6 + 7 = 4

29 -3.8

105, 85

650 -606

38 seats; 324 seats

3072 8192

10

63_16

208.32

$1628.89; $13,206.79

400_3 or 133.

− 3 14

916 Chapter 12

C H A P T E R

12

OrganizerObjective: Assess students’ mastery of concepts and skills in Chapter 12.

Online Edition


Chapter 12 Tests

• Free Response (Levels A, B, C)

• Multiple Choice (Levels A, B, C)

• Performance Assessment

IDEA Works!

Modified Chapter 12 Test

Answers

21. 2 (1) -1

_ 2 = 1 2 _

2 = 1 _

2

22. 1 _ 2 + + 2k - 1 _

2 = k 2 _

2

23. 1 _ 2 + + 2k - 1 _

2 +

2 (k + 1) - 1 __

2

= k 2 _ 2 +

2 (k + 1) - 1 __

2

= k 2 + 2k + 2 - 1 __

2

= k 2 + 2k + 1 __

2 =

(k + 1) 2 _

2

College Entrance Exam Practice 917

FOCUS ON SATWhen you get your SAT scores, you are given the percentile in which your scores fall. This tells you the percentage of students that scored lower than you did on the same test. You’ll see your percentile score at the national and state levels. They are usually not the same.

You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete.

1. The first term of a sequence is 6, and each successive term is 3 less than twice the preceding term. What is the sum of the first four terms of the sequence?

(A) 27

(B) 30

(C) 51

(D) 57

(E) 123

2. The first term of a sequence is 2, and the nth term is defined to be 3n - 1. What is the average of the 7th, 10th, and 12th terms?

(A) 24.5

(B) 28

(C) 29

(D) 32

(E) 84

3. The first term of an arithmetic sequence is -5.If the common difference is 4, what is the 7th term of the sequence?

(A) -20,480

(B) -29

(C) 19

(D) 20

(E) 23

4. A population of 50 grows exponentially by doubling every 4 years. After how many years will the population have 1600 members?

(A) 20

(B) 16

(C) 10

(D) 6

(E) 5

5. Which of the following sequences can be expressed by the rule an = n - 1____

n + 1 ?

(A) 3, 2, 5_3

, 3_2

, 7_5

, . . .

(B) 1_2

, 3_4

, 5_6

, 7_8

, 9_10

, . . .

(C) 1_3

, 2_4

, 3_5

, 4_6

, 5_7

, . . .

(D) 0, 1_3

, 1_2

, 3_5

, 2_3

, . . .

(E) 0, 3_2

, 3, 5_3

, 5, . . .

6. Which of the following sequences is a geometric sequence?

(A) -7, 14, -28, 56, -112, . . .

(B) -4, -6, -8, -10, -12, . . .

(C) -3, 1, -3, 1, -3, . . .

(D) 4, 12, 48, 144, 576, . . .

(E) 1, 4, 9, 16, 25, . . .

Read each problem carefully, and make sure that you understand what the question is asking. Before marking your final answer on the answer sheet, check that your answer makes sense in the context of the question.

College Entrance Exam Practice 917

OrganizerObjective: Provide practice for college entrance exams such as the SAT.

Online Edition

ResourcesCollege Entrance Exam Practice

Questions on the SAT represent the following math content areas:

Number and Operation, 30–32%

Algebra and Functions, 28–32%

Geometry and Measurement, 27–30%

Data Analysis, Statistics, and Probability, 10–12%

Items on this page focus on:

• Number and Operation

• Algebra and Functions

Text References:

Item 1 2 3 4 5 6

Lesson 12-2 12-1 12-3 12-4 12-1 12-4

C H A P T E R

12

1. Students may choose A because it is the fourth term of the sequence. Students may choose B because it is the sum of the first three terms. Remind students to read each question carefully.

2. Students may spend more time than is necessary on this problem if they find all the terms of the sequence from the 1st term through the 12th term. Encourage students to use a formula to find the 3 terms. Then remind them to find the average of these terms.

3. Students may choose B because they subtracted 4 from each successive term instead of adding 4 to each successive term. Students may choose E because it is the 7th term of the sequence after the first term.

4. Students may choose D because the population is 1600 after 5 doubling periods. Remind them that each period represents 4 years.

5. Students may choose A because it is close to the sequence given by the rule n + 1 _____ n - 1 . Students may choose C because it is the correct sequence but missing the first term.

6. Students may choose B if they do not know the difference between arithmetic and geometric sequences. Students may choose D because alternating terms have a common ratio.

Simple interest formula


Short Response Ariana is saving money for a new car. She saves $40 the first week, $45 the second week, $50 the third week, and so on. Explain whether an arithmetic or geometric sequence would best represent this situation. Use a sequence or series to determine the amount that she will save in the 8th week and the total amount that she will have saved after 8 weeks.

Create an outline for your response.

Follow the outline, and write out your response.

Short/Extended Response: Outline Your ResponseAnswering short and extended response items on tests is a lot like writing essays in English class. You can use an outline to plan your response to the question. Outlines help you organize the main points and the order in which they will appear in your answer. Outlining your response will help ensure that your explanation is clearly organized and includes all necessary information.

An arithmetic sequence would best represent this situation because Ariana is adding $5 each week to the amount that she saves. This would be an arithmetic sequence where the first term is 40 and the common difference is 5.

The sequence for the amount saved each week is an = 40 + 5 (n - 1) . The series for the total amount

saved is ∑

k = 1

n

[40 + 5 (k - 1)] .

The amount saved in the 8th week is a8 = 40 + 5 (8 - 1) = $75.

The total saved after 8 weeks is ∑

k = 1

8

[40 + 5 (k - 1)] = $460.

Clearly indicate which is the sequence and which is the series for the second step.

Include evidence to explain the answer for the first step.

Show how you found the answers for the last two steps.

Outline

1. Explain whether arithmetic or geometric.2. Write sequence and series.3. Find amount saved in 8th week.4. Find total saved after 8 weeks.

918 Chapter 12

C H A P T E R

12

OrganizerObjective: Provide opportunities to learn and practice common test-taking strategies.

Online Edition

This Strategies for Success shows the benefits of outlin-

ing a response to either a short- or extended-response test item. Advise students that outlining a response to a multipart test item is similar to writing an outline for an essay in an English or history class. Remind stu-dents to include an answer check or verification of units as a step in the outline.

Encourage students to use short, concise statements or sentences in their outlines. Tell students that an outline is not just a restatement of the problem, but that it should include details of a plan for solving and explaining each step.

Strategies for Success 919

When you finish your response, check it against your outline to make sure that you did not leave out any details.

Read each test item and answer the questions that follow.

Item AShort Response Explain how to determine whether an infinite geometric series has a sum.

1. What should be included in an outline of the response for this test item?

2. Read the two different outlines below. Which outline is the most useful? Why?

Student A I. Definition of an infinite geometric series

and the common ratio r.

II. Definition of the sum of an infinite geometric series.

III. Explain for which values of r that a sum exists.

Student BA. Geometric series has a common ratio.

B. Common ratio has to be less than 1.

Item B Extended Response A pattern for stacking cereal boxes is shown at right.a. Explain how many

boxes are in a 9-row display.

b. If 91 boxes are to be stacked in this display, explain how many rows the display will have.

3. Read the outline below. Identify any areas that need improvement. Rewrite the outline to make it more useful.

Outline 1. Find the number of boxes if there are

9 rows.

2. Find the number of rows needed for 91 boxes.

Item C Extended Response A pattern of squares is created by doubling the dimensions of the previous square. Explain how to find the sum of the perimeters of the first 8 squares if the first square is 5 cm wide.

4. A student correctly gave the following response. Write an outline for a response to this question.

To find the perimeters of the first 8 squares, I need to determine the series.

4(5) + 4 (10) + 4 (20) + 4 (40) + �

4(5 + 10 + 20 + 40 + �)

4 · 5 (1 + 2 + 4 + 8 + �)

20(1 + 2 + 4 + 8 + �)

20∑

n = 1

8

2(n-1)

Now that I know that the first term is 20 and the common ratio is 2, I can use the formula

Sn = t1(

1 - rn_1 - r )

, where t1 is the first term

and r is the common ratio.

Sn = 20 (1 - 28_1 - 2 )

= 20 (

1 - 256_-1 )

= 20 (

-255_-1 )

= 20 (255) = 5100

So the sum of the perimeters of the first 8 squares is 5100 cm.

AnswersPossible answers:

1. Outlines should define an infi-nite geometric series and com-mon ratio, define the sum of an infinite geometric series, and explain how the common ratio affects whether or not the series has a sum.

2. Student A’s outline is most useful because it will guide the student to a more complete answer.

3. The outline only restates the directions. Improved outline:

I. Identify the sequence by find-ing the differences between terms.

II. Use the formula for the nth term of an arithmetic sequence to find the number of boxes in a 9-row display.

III. Use the sum formula for an arithmetic series to find the number terms that have a sum of 91.

4. I. Find the perimeters of the first 4 squares.

II. Write the sum of these perimeters as a series.

III. Determine the first term of the series and the common ratio.

IV. Use the formula

S n = a 1 ( 1 - r n _ 1 - r

) , where a 1 is

the first term and r is the common ratio, to determine the sum of the perimeters of the first 8 squares.

Answers to Test ItemsSee answers to related problems.

Strategies for Success 919


CUMULATIVE ASSESSMENT, CHAPTERS 1–12

Multiple Choice

1. Which shows the series in summation notation?

4 + 6 + 4 + 6 + 4

∑24

∑

n = 0

5

[(-1)

n+ 5]

∑

n = 1

4

[(-1)

n+ 5 ]

∑

n = 1

5

[(-1)

n+ 5 ]

2. What is the expanded binomial?

(2x - y)

3

x3- 3 x2y + 3xy2

- y3

8x3- 12 x2y + 6xy2

- y3

x3+ 3 x2y + 3xy2

+ y3

8x3+ 12 x2y + 6xy2

+ y3

3. Let f (x) = x3+ 2 x2

- 5x - 9. Which function would show f (x) reflected across the y-axis?

g(x) = - x3- 2 x2

+ 5x + 9

g(x) = - x3+ 2 x2

+ 5x - 9

g(x) = 2 x3+ 4 x2

- 10x - 18

g(x) = x3+ 2 x2

- 5x - 5

4. Which function shows exponential decay?

f (x) = -5x

f (x) = 2.3(6.7)

x

f (x) = 0.49(7.9)

x

f (x) = 5.13(0.32)

x

5. A ball is dropped from a height of 10 feet. On each bounce, the ball bounces 60% of the height of the previous bounce. Which expression represents the height in feet of the ball on the nth bounce?

10 (0.6n)

10(0.6)

n-1

10 - n_0.6

10(0.6)

n

6. Which is the graph of the inequality 6x + 3y ≥ 9 x2

- 3?

7. Gina opened a new deli. Her revenues for the first 4 weeks were $2000, $2400, $2880, and $3456. If the trend continues, which is the best estimate of Gina’s revenues in the 6th week?

$3856

$4032

$4147

$4980

8. What is the 9th term in the sequence?

an =

1_2

(2n-1) + 4

36 132

68 260

9. Find the inverse of f (x) = 4x - 5.

f-1(x) = -4x + 5

f-1(x) =

1_4

x + 5

f-1(x) =

x + 5_4

f-1(x) = 5x - 4

KEYWORD: MB7 TestPrep

920 Chapter 12

C H A P T E R

12

OrganizerObjective: Provide review and practice for Chapters 1–12 and standardized tests.

Online Edition


Chapter 12 Cumulative Test

Focus on CAHSEE Standards: Benchmark Tests and Intervention

California Standards Practice CD-ROM

KEYWORD: MB7 TestPrep

For Item 13, students will often forget that one of the 10 cards is also a diamond. Remind them that they must account for this by subtracting the probability of this one card from the combined probability.

In Item 11, you may choose to graph, factor, complete the square, or use the Quadratic Formula to find the zeros.

10. What transformation has been applied to f to get g?

Horizontal compression by 1_5

Horizontal stretch by 5

Vertical compression by 1_3

Vertical stretch by 5

Cumulative Assessment, Chapters 1–12 921

Short Response16. Use the function f (x) =

3√ � 5x to answer the

following questions.

a. What is the domain and range?

b. What is the inverse of f (x)?

c. What is the domain and range of the inverse function?

d. Graph f (x) and f-1(x) on the same coordinate

plane.

17. Use the infinite geometric series ∑

n = 1

∞

5____4n - 1 to

answer the following questions.

a. Determine if the series converges or diverges.

b. Find the sum of the infinite series, if it exists.

18. A grocery store display contains 3 cans on the top row and an additional can in each row forming a triangular shape.

a. Would you use a sequence or a series to represent the number of cans in the nth row? Explain.

b. How many cans are in the 12th row?

c. What does the series ∑

k = 1

n

(k + 2) represent? Explain.

Extended Response19. A test to be on a trivia show has two parts. 60%

of contestants pass the first part, and 20% pass the second part.

a. Draw a tree diagram that gives the probabilities for a contestant’s possible outcomes on the test.

b. If a contestant must pass both parts of the test to be on the show, how many contestants out of a group of 50 would likely make the show? Show your work.

c. Is it more likely that a contestant would pass both parts or fail both parts of the test? Explain.

11. Find the zeros of f(x) = 2 x2+ 5x - 12.

-4, 3_2

-2, 3

-

3_2

, 4

3_2

, 2

Gridded Response12. Find the common ratio of the geometric

sequence.

125, 50, 20, 8, . . .

13. A card is drawn from a deck of 52. What is the probability of drawing a 10 or a diamond, to the nearest hundredth?

14. What is the sum of the arithmetic series?

∑

k = 1

8(7k - 3)

15. What is the y-value of the point that represents the solution to the given system of equations, to the nearest hundredth?

⎧

⎨

⎩

2y - 2 = 4x

6 - x = 8y

0.4

0.31

228

0.76

D: �; R: �

f -1(x) = x3_

5D: �; R: �

converges

20_3

Cumulative Assessment 921

Short-Response RubricItems 16–18

2 Points = The student’s answer is an accurate and complete execu-tion of the task or tasks.

1 Point = The student’s answer con-tains attributes of an appropriate response but is flawed.

0 Points = The student’s answer contains no attributes of an appropriate response.

Extended-Response RubricItem 19

4 Points = The student correctly draws and labels a tree diagram in part a, correctly answers part b, answers and explains part c, and shows all work.

3 Points = The student correctly draws and labels a tree diagram in part a and answers parts b and c correctly, but does not show all work.

2 Points = The student correctly draws and labels a tree diagram in part a but does not correctly answer either part b or c.

1 Point = The student draws an incomplete tree diagram in part a and does not correctly answer parts b and c.

0 Points = The student does not answer correctly and does not attempt all parts of the problem.

Answers 16d.

18a. a sequence because each term in the sequence represents the number of cans in each row

b. 14

c. the total number of cans in n rows 19a.

b. 6; the probability of passing both parts is 0.6 (0.2) = 0.12. Out of 50 contes-tants, 0.12 (50) = 6 would most likely pass.

c. It is more likely that a contestant fails both parts of the test. The probability of passing both parts is 12%, but the probability of failing both parts is 32%.


N E V A D A

Hoover Dam

The Hoover DamSince its completion in 1935, the Hoover Dam has often been cited as one of the seven engineering wonders of the world. Its 6.6 million tons of concrete tame the waters of the Colorado River and form Lake Mead, the largest man-made reservoir in the United States.

Choose one or more strategies to solve each problem. For 1 and 2, use the table.

Traffic Forecasts for the Hoover Dam

Year 2007 2008 2009

Number of Cars per Day 16,300 16,780 17,260

1. The Hoover Dam serves as a bridge between the Nevada and Arizona sides of the Colorado River. Traffic analysts project that the traffic on the dam will increase according to an arithmetic sequence. How many cars, on average, would you predict to cross the dam each day in 2017?

2. Approximately how many vehicles will cross the dam in the years 2007 through 2017, inclusive? (Hint: Assume 365 days per year.)

3. Small trucks make up 18% of the traffic on the dam, and RVs account for another 4%. All trucks and RVs are inspected before they are allowed to cross.

a. Assuming that other types of vehicles are not inspected, what is the probability that three consecutive vehicles arriving at a checkpoint will be inspected?

b. What is the probability that 3 out of 5 vehicles arriving at a checkpoint will need to be inspected?

21,100

about 75 million

0.01

0.06

Problem Solving on Location

OrganizerObjective: Choose appropriate problem-solving strategies and use them with skills from Chapters 11 and 12 to solve real-world problems.

Online Edition

The Hoover Dam

Reading StrategiesAs students read Problem 1, check that they understand the words forecast and projection. If students are unfamiliar with these terms, ask them to reread the problem and use context clues to guess the definitions.

Using Data Ask students how they could estimate the total number of cars crossing the Hoover Dam in 2007. Multiply 16,300 by 365.

Problem-Solving FocusAsk students what strategy they would use to solve Problem 2. Students may wish to use the strategy Solve a Simpler Problem and/or Make an Organized List. Finding the total number of cars for one year and multiplying by 11 will give students a good estimate to compare with their final answers.

922 Chapter 12


Problem Solving on Location 923

Problem SolvingStrategies

Draw a DiagramMake a ModelGuess and TestWork BackwardFind a PatternMake a TableSolve a Simpler ProblemUse Logical ReasoningUse a Venn DiagramMake an Organized List

Silver and Gold MiningNevada’s nickname is the Silver State, which is not surprising considering that more than 10 million ounces of the metal are mined in Nevada each year. Gold mining is also an essential part of Nevada’s economy. In fact, if Nevada were a nation, it would rank third in the world in gold production behind South Africa and Australia.

Choose one or more strategies to solve each problem.

1. Nevada experienced a gold-mining boom from 1981 to 1990. During this period, the number of thousands of ounces of gold mined each year can be modeled by a geometric sequence in which a1 = 375 (that is, 375,000 ounces were mined in 1981) and the common ratio is 1.35. Approximately how many ounces of gold were mined in 1990?

2. What was the total gold production in the years 1981 through 1990, inclusive?

3. In a particular mine, the probability of discovering a profitable quantity of gold in a sector is approximately 40%. What is the probability that the miners will discover a profitable quantity of gold in 3 of the next 4 sectors?

For 4, use the table.

4. Nevada experienced a silver boom during the 1990s. An industry analyst is collecting detailed data for all of the years from 1991 to 2000 in which silver production was outside of 1 standard deviation of the mean. For which years should she collect this data?

Nevada Silver Production

YearProduction (million oz)

1991 18.6

1992 19.7

1993 23.2

1994 22.8

1995 24.6

1996 20.7

1997 24.7

1998 21.5

1999 19.5

2000 23.2

5,585,000 oz

20,471,000 oz

15.36%

1991, 1992, 1995, 1997, and 1999

Problem Solving on Location 923

Silver and Gold Mining

Reading Strategies


Check that ELL students are familiar with the word mining. Students may recognize similarities to words in other languages. For example, the Spanish word for miner is minero.

Using Data In discussing the data presented in Problem 1, ask students to describe the annual rate of increase in gold production as a percent. 35% Before students do any calculations, ask them to make some ballpark predictions about gold production in 1990.

Problem-Solving FocusAlthough students can use formulas to solve Problems 1 and 2, they may gain a deeper understanding of these problems if they also explore them using problem-solving strategies. For example, as part of the final step of the problem-solving process, (4) Look Back, ask students which strategy might be useful in checking the answers to these problems. Possible answer: Use the strategy Make a Table. To check the answer to Problem 1, look at the gold production for 1990 in the table. To check the answer to Problem 2, add all the values in the production column of the table.

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