Alg 2H ASSIGNMENT SHEET (No applications)

Chapter 11: Sequences and Series / Final Exam Review #1 a) 11-2 Complete: 11-2 Arithmetic and Geometric Sequences Lesson Worksheet #1 b) 11-2 p.570 3, 7, 9, 11, 12-45 multiples of 3 c) Final Rev Review for Final Exam Worksheet A: 1, 2 #2 a) 11-3 Complete: 11-3 Arithmetic and Geometric Means Lesson Worksheet #2 b) 11-3 p.575 1, 7-27odd, 31 c)Review p.570 5, 13, 16, 22, 29, 35, 40, 43, 46 d) 11-3 Complete the following problems: (1) For the arithmetic sequence: Given: t7= -13, t21= -9 ½ . Find t100 (Ans: d= ¼ , t100=10 ¼ ) (2) For the geometric sequence: Given: t9= -30,000, t19=-.003072. Find t25 (Find real solutions only) (Ans: r=.2 or -.2 t25=-1.96608 x 10-7) e) Final Rev Review for Final Exam Worksheet A: 3 #3 a) 11-4 Complete: 11-4 Introduction to Series—Sigma Notation Lesson Worksheet #4 b) 11-4 p.579 1, 5, 7, 15, 17, 19 c) Review 11-2,11-3 Sequences and Means Practice Worksheet #3: 1-27odd d) Final Rev Review for Final Exam Worksheet A: 4 #4 a) 11-5 Complete: 11-5 Series Lesson Worksheet #5 b) 11-5 11-4, 11-5 Sequences and Series Worksheet #6: 3, 5, 7(Arithmetic),11, 13, 21, 22, 29 c) 11-5 p.588 27, 31 d) Review p.580 9, 11 e) Final Rev Review for Final Exam Worksheet A: 5 #5 a) 11-5 Complete: 11-5 Series using Sigma Notation Lesson Worksheet #7a b) Review 11-4, 11-5 Sequences and Series Worksheet #6 : 1, 9(Arithmetic), 15, 17(Geometric), 25, 27, 30(Answer: 16 days later) c) 11-5 p.588 28(Answer: 303) , 32 (Answer: 25)

d) Final Rev Review for Final Exam Worksheet A: 6 #6 a) 11-5 Complete: 11-5 Series using Sigma Notation Lesson Worksheet #7b b) 11-6 Complete: 11-6 Convergent Geometric Series Lesson Worksheet #9 c) 11-6 p. 595 3, 5, 9 d) Review 11:2 – 11:5 Review Worksheet #8: Multiples of 3 e) Review all past Ch11 homework. Check all answers. Refer to solutions where possible.

#7 a) Review Complete: 11:2 – 11:5 Review Worksheet #8

b) ChRev Review for Chapter 11 Test Worksheet #10: Multiples of 3 (omit 31-33) b) Final Rev Review for Final Exam Worksheet A: 7, 8 Check all answers. Refer to solutions where possible. Due Day of Chapter 11 Test:

#8 a) ChRev Review for Chapter 11 Test Worksheet #10: all (omit 31-33) b) Final Rev Review for Final Exam Worksheet A: 9, 10

BE READY FOR POP-QUIZZES AT ANY TIME DURING THIS CHAPTER

Final Exam Review:

Complete all review problems on separate paper. Show all work!!! Do not mark answers on the Review Packets so you can complete them again before the exam.

NOTE ANY PROBLEMS YOU DID INCORRECTLY. BE SURE TO REVIEW ALL OF THEM AGAIN BEFORE THE EXAM!

Due Day after Chapter 11 Test:

#9a) Final Rev Complete: Review for Final Exam Worksheet A b) Final Rev Multiple Choice Final Exam Review: 7, 22, 32, 59, 67, 89, 102 #10 a) Final Rev Multiple Choice Final Exam Review: 7, 15, 28, 32, 36, 46, 50, 57, 59, 70, 79, 86, 100, 101 b) Final Rev Review for Final Exam #2: Multiples of 3 #11 a) Final Rev Multiple Choice Final Exam Review: evens b) Final Rev Review for Final Exam #2: evens

#12 a) Final Rev Multiple Choice Final Exam Review: Complete all b) Final Rev Review for Final Exam #2: Complete all

It is essential that you complete and study ALL the review problems carefully. You need to know how to do ALL of them in order to succeed on this challenging final exam.

WK #1 Alg2H Date ________

11-1, 11-2 Arithmetic and Geometric Sequences Lesson Worksheet #1

Definition: Examples: 1) A Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, . . . . . . . . . . Pattern: _______________________________________________________________ Find t8 , the value of the 8th term.: _______

(This could be written as t(8) , the value of the term when n=8, in function notation.) 2) The sequence of triangular numbers: 1, 3, 6, 10, 15, 21, 28, . . . . . . . . . Pattern: _______________________________________________________________ Find t8.: _______ 3) An Arithmetic sequence: 3, 6, 9, . . . . . . . . . Pattern: _______________________________________________________________ Find t4:________ Definition:

For the arithmetic sequence: 2, 5, 8, 11 . . . . . . Find t20 = _______________________________________________________________ Formula:

Sequence: A set of numbers that follows a pattern. A function whose domain, the term numbers (n), is the set of natural numbers

and whose range is the set of term values (tn).

Arithmetic Sequence: A sequence in which one terms equals a constant ________________ the preceding term.

The constant is called the common _______________________.

For an Arithmetic Sequence: tn = ___________________________________ In words: The nth terms equals the ________ term added to (________) common differences.

This formula is an equation for a ___________________ function.

4) A Geometric Sequence: 3, 6, 12, . . . . . . Pattern: _______________________________________________________________ Find t4:________ Definition:

For the geometric sequence: 3, 6, 12, . . . . . . .

Find t20 = _______________________________________________________________

Formula: 11-2 Sample problems: 1) Determine if the following sequences are Arithmetic, Geometric or neither. If Arithmetic or Geometric, determine the 31st term. (If neither, 31st term not required) a) 7, -4, -15, . . . . . . . Type: ________________ t31 = _____________________________________________________________________ b) Type: ________________ t31 = _____________________________________________________________________

c) ....7,7,7 63 Type: ________________ t31 = _____________________________________________________________________ 2) What term has the value of 34816 in the geometric sequence 17, 34 . . . . . . . . . . . . . . 34816? (What term is 34816 in the Geometric sequence with t1 =17, and common ratio, r = 2?)

Geometric Sequence: A sequence in which one term equals a constant _________________the preceding term.

The constant is called the common _______________________.

For an Geometric Sequence: tn = ___________________________________ In words: The nth terms equals the ________ term multiplied by (________) common ratios.

This formula is an equation for a ___________________ function.

12

13

29

, , ,.......

3) Determine the number of terms in the Arithmetic Sequence: 305, 297, . . . . . . . 17 Find out which term the given number is in the indicated sequence:

4) – 1953125 in the sequence 251 ,

51

− , 1 , ……………….(Technique to deal with negative common ratios)

5) 305 in the sequence 17, 25, 33...... ⇒over

Find out which term the given number is in the indicated sequence: 6) Approximately 2270.6051 in the sequence 17, 18.02, 19.1012 ....

Alg2H Name __________________ Date ___________WK#2 11-3 Arithmetic and Geometric Means Lesson Worksheet #2 Definition: Sample Problems: 1) Find 4 arithmetic means between 51 and 37. 2) Find one arithmetic mean between 10 and 20. (Find the arithmetic mean of 10 and 20) (Note: Inserting one arithmetic mean midway between the two given numbers is the same as

determining the midpoint or “average”.) 3) Find one geometric mean between 4 and 16. (Find the geometric mean of 4 and 16) (Solve resulting quadratic by factoring)

Arithmetic Means between two numbers: Numbers which form an arithmetic sequence with the two given numbers.

Geometric Means between two numbers: Numbers which form a geometric sequence with the two given numbers.

4) Find 3 geometric means between 6 and 96, allowing the common ratio to be a complex number. (There may be more than one set) For r = ____ : 6, _____, ______, _______, 96 For r = ____ : 6, _____, ______, _______, 96 For r = ____ : 6, _____, ______, _______, 96 For r = ____ : 6, _____, ______, _______, 96 5) Given an arithmetic sequence with t10 = 7 and t19 = 10. Find t1 and t40 6) Find 2 geometric means between 1/9 and –3 allowing the common ratio to be a complex number. (There may be more than one set)

Alg2H Name ________________ Date _______ WK#4 11-4 Introduction to Series—Sigma Notation Lesson Worksheet#4 Definition: Arithmetic Sequence; 9, 5, 1, -3, . . . . . . . . . Arithmetic Series: 9 + 5 + 1+ -3 + . . . . . . . . Sigma notation Read: “The sum from k = 1 to k = n of tk.” The k is called the index. (It’s a counter) Sample Problems: 1) Evaluate the expression by writing the terms and adding them up: a) b) c)

2) For each problem, write Sn using sigma notation. a) S5 for 9 + 5 + 1+ -3 + . . . . . . . . b) S100 for

Series: An indicated sum of the terms of a sequence.

Sn, nth partial sum of a series: Sum of the first n terms of that sequence.

S tn kk

n

= ==∑

1

2 51

3

+=∑ kk

2 3 1

1

3

• −

=∑ k

kc h

2 3 1

1

3

• −

=∑ k

kc h

13

16

112

+ −FHGIKJ +FHGIKJ+.........

Alg2H 11-5 Series Lesson Worksheet #5 Date __________ WK#5 pg. 1 of 3 Definitions: Partial Sum of an Arithmetic Series Karl Friedrich Gauss, a famous 19th Century mathematician , was a child prodigy. It is said that when Gauss was ten his teacher asked his class to add the numbers from 1 to 100 to keep them busy and quiet for a long time. The teacher was astonished when Gauss quickly found the answer by mentally figuring the summation. His technique:

1 + 2 + 3 + 4 + . . . . . . . . . . . . . . . . . . . . . .97 + 98 + 99 + 100 + Formula:

Sample Problem: 1) Find the sum of the first 50 positive 2) Given the sequence: 14 + 12 + 10 + . . . . multiples of 7. If Sn = -54, find n (the # of terms in the partial sum)

Arithmetic Series: an indicated sum of the terms of an arithmetic sequence Geometric Series: an indicated sum of the terms of an geometric sequence

The nth Partial Sum of an Arithmetic Series: Sn = In words:

Partial Sum of a Geometric Series WK#5 pg. 2 of 3 Develop a rule for Sn:

Sn = t1 + t1r + t1r2 + t1r3 + . . . . . . + t1rn-1 -rSn = - t1r - t1r2 - t1r3 - . . . . . . - t1rn-1 – t1rn (Multiply each term in the series by –r) Sn-rSn = (Add) (Factor both sides) (Solve for Sn) Formula:

Sample Problems: 1) You Mother Father Mother Father Mother Father Mother Father Mother Father Mother Father Mother Father You decide to make a family tree of you, your parents, grandparents, great-grandparents etc that preceded you. How many people will be on your family tree if it covers 8 generations, including you? 2) Determine the sum of the series: 5 + 35 + 245 + . . . . . . . 4117715

The nth Partial Sum of a Geometric Series with common ratio r ≠ 1: Sn =

WK#5 pg. 3 of 3 3) Given: 7 + 7.42 + 7.8652 + . . . . . Find the number of terms whose sum is approximately 3308.5414 4) Sn=26,361 in an arithmetic series 2 + 9 + 16 + 23 +......... Find n.

Alg2H 11-5 Series using Sigma Notation Lesson/HW Worksheet #7a Date ______WK #7a 1) Write in Sigma Notation. 2) Then evaluate each sum using the appropriate Sum formula 1. 5 + 8 + 11 + ……+ 65

2. 31250

1.....501

101

21

++−

3. 4 + 5 + 6 + 7 + ….. + 23

4. For the geometric series t3 = 1, t5 = 94 . Write S10 using sigma notation and evaluate the sum.

(Remember: In this problem there are 2 sets of answers because there are 2 possible values for r ) 5. In the geometric series t7 = 4 and t11 = .0004. a) Find r, allowing r to be complex b) Find t8, t9, t10 for each value of r c) For the positive real value of r , Sn = 4,444,444. Find n

Alg2H 11-5 Series using Sigma Notation HW (WK#7b) Date ______ WK #7b 1) Write in Sigma Notation. 2) Then evaluate each sum using the appropriate sum formula.. 1. 1 + 5 + 9 + 13 + 17

2. 3072....323

43

+++

3. 2 + 4 + 8 + 16 + 32 + 64

4. 6 – 36 + 216 ……….-60466176 5. 8 + 13 + 18 + ….. If Sn = 4859, find n

Alg2H 11:2-11:5 Review Worksheet #8 Date ________ WK #8 Complete on a separate paper! Copy each problem and show all work. State each formula used. 1. Find a56 of the sequence 237, 230, 223….. 2. Find the number of terms in the sequence 164, 155, 146, ……-484 3. Find the first term of the arithmetic sequence with a6 = 17 and a15 =217 4. Determine three arithmetic means between 3 and 48. 5. To 4 decimal places determine a30 in the sequence 17, 20.4, 24.48 ……. 6. Determine n if a1 = 11, r = 3 and an = 24057. 7. In a geometric sequence, a4 = 2000 and a7 = 250. Write Sn using sigma notation if r is real. 8. Determine three geometric means between 3 and 48 if r (common ratio) can be complex. 9. Write S13 using sigma notation and determine S13 of an arithmetic series if a7 = 3 and a13 = 51.

10. Find the sum of the first 27 terms of the series: ..........314

213

322 ++

11. Write using sigma notation. Then evaluate the sum: 1 + 3 + 9 +……..+2187 12. Given: Arithmetic series, S348 = 3534, d = 5. Find t20 13. Given: Geometric series, t1 = 20, t3 = 18.05.

Write S18 in sigma notation and evaluate the sum, rounded to 4 decimal places, if r is real 14. Given: Arithmetic series, t5 = 138, t8 = 133.5, Sn = 3667.5. Determine n. 15. a) Determine the number of multiples of 6 between -32 and 601. b) Find the sum of the multiples of 6 between -32 and 601

ANSWERS: 1. a56 = -148 2. n = 73

3. a1 = -9491

4. 3, 14.25, 25.5, 36.75, 48 5. a30 = 3362.8311 6. n = 8

7. ∑=

−n

k

k

1

1)5(.000,16

8. r = 2 : 3, 6, 12, 24, 48 r =-2 : 3, -6, 12, -24, 48 r = 2i : 3, 6i, -12, -24i, 48 r =-2i : 3, -6i, -12, 24i, 48

9. S13 = )538(13

1∑=

−k

k = 39

10. S27 = 36421

11. S8 = ∑=

−8

1

1)3)(1(k

k = 3280

12. t20 = 3448276.76229

22108-−=

13. S18 = ∑=

−18

1

1)95)(.20(k

k = 241.1143 or S18 = ∑=

−−18

1

1)95.)(20(k

k = 6.1824

14. n = 30 or 163 15. a) n = 106 (For: -30, -24, …….600) b) S106 = 30210

Alg2H 11-6 Convergent Geometric Series Lesson Worksheet #9 Date ______ WK#9 1. Determine r and Sn for each geometric series:

(1) .....121

41

++ (2) 4 + 2 + 1……

r = ________ r = _________ Sn = Sn = 2. Enter each expression for Sn in TI-83 calculator in y1 and y2 , respectively. 3. Go to TBLSET. Set start = 1, ∆Tbl = 1, Indpnt: Auto, Depend: Auto 4. Go to Table and scroll down. a. For problem (1) where r = 2, what happens to Sn as n gets larger and larger? _____________________________________________________________________

b. For problem (2) where r = 21 , what happens to Sn as n gets larger and larger?

_____________________________________________________________________ 5. Set WINDOW: domain [0, 10] and range [0,10]. View the graphs of each of these sets of partial sums. Which Sn approaches a horizontal asymptote as n ⇒ ∞ __________________ This represents a CONVERGENT SERIES. Definition: A series converges to a number S if the partial sums, Sn, stay arbitrarily close to S as n gets very large. For the convergent series in this example, S = _________

(the same as the y value of the horizontal asymptote)

Do the partial sums, Sn, in the other example stay close to any number as n gets very large? _______________________

6. Compare the two formulas for the partial sums, Sn, to understand why.

(1) For r = 2 (2) For r = 21

lim n⇒∞ Sn = )21()21(

41

−− n

lim n⇒∞ Sn = )

211(

)211(

4−

⎟⎠⎞

⎜⎝⎛−

•

n

7. In general when will a series converge?

S = lim n⇒∞ Sn = =−−

)1()1(

1 rrt

n

CONVERGENT GEOMETRIC SERIES: A Geometric series converges if ___________________________________ The limit, S, to which it converges is given by S = lim n⇒∞ Sn = 8. Question: Could an infinite arithmetic series converge? lim n⇒∞ Sn =

Determine whether or not each series converges. If so, find the value(s) to which it converges.

1. Geometric series with t1 = 60 t3 = 632 (Remember 2 answers!)

2. 42 – 10.5 + 2.625 …….. Over ⇒

3. ........031

32

++ …….. 4. 3.2 + 4 + 5………

5. Geometric series with t2 = 27 and t6 = 31 (for real values of r) (Remember 2 answers!)