algebra ii unit 8 1 - lsrhs.net 8.pdf · algebra ii unit 8 1. use the rule to write the first three...

The table shows the cost of mailing letters first class in the U.S. in 1995.

Weight Not Exceeding (ounces)

Cost

1 $0.32

2 $0.55

3 $0.78

4 $1.01

5 $1.24

$0.32, $0.55, $0.78, $1.01, $1.24 This set of numbers is an example of a sequence.

Each number in the sequence is called a term.

The first term is symbolized by a1 , the second term by a2 , and so on to an , the nth term.

a1 = $0.32 a2 = $0.55 a3 = $0.78 a4 = $1.01 a5 = $1.24

You can find the next term in a sequence by looking for a pattern.

Example: Given the sequence: 3, 5, 7, 9, 11, ... A. Find the next 3 terms of the sequence.

13, 15, 17

B. Find the rule for the nth term.

1 2 3 4 5 ••• n3 5 7 9 11 ••• an

Notice each number in the sequence 3, 5, 7, 9, 11 is equal to one more than twice the positive integer with which it is paired.

Rule: an = 2n + 1

Algebra II Unit 8 1

Use the rule to write the first three terms of each sequence:

1. an = 5n

2. an = n +

1n

3. an = n2 +1

4.

€

an = −4n2 − 2

Find the next 3 terms of the sequence.Find the rule for the nth term.

5. 2, 4, 6, 8, ...

6. 4, 7, 10, 13, ...

7. 48, 45, 42, 39, ...

8. 2, 5, 8, 11, ...

9. -12, -7, -2 , 3, ...

10. 12

, 23

, 34

, 45

,...

Algebra II Unit 8 2

Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between successive terms is a constant.

The constant is called the common difference, d.

Formula:

An arithmetic sequence: -6, -2, 2, 6, 10, ... common difference is 4.

Example 1: The first term of an arithmetic sequence is 3 and the common difference is 2. Find the twentieth term.

an = a1 + (n −1)d

a20 = 3 + (20 − 1)2

a20 = 3 + (19)2

a20 = 3 + 38a20 = 41

1. Write the first five terms of each arithmetic sequence:

A. a1 =1, d=5

B. a1 =-5, d=6

C. a1 =-30, d=-2

D. a1= 34

, d= −14

Algebra II Unit 8 3

2. The first term of an arithmetic sequence is 16. The common difference is -2. Find a17.

3. The first term of an arithmetic sequence is 2. The common difference is 5. Find a6 .

4. In an arithmetic sequence a1 = 20 and d =

52

. Find a11.

5. In an arithmetic sequence a1 = −6 and d =

23

. Find a10 .

6. The first term of an arithmetic sequence is 2 and the common difference is 3. find the 30th term.

Algebra II Unit 8 4

7. The first term of an arithmetic sequence is 5 and the common difference is -4. find the twenty-fifth term.

8. Find the 10th term of 4, 5, 6, ...



11. The top row of a pile of logs contains 6 logs, the row below the top one contains 7 logs, the third row from the top contains 8 logs, and so on. If there are 45 rows, how may logs are there in the bottom row?

Algebra II Unit 8 5

Example 2: The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1

and d.

___ ___ ___ ___ ___ - 2 ___ ___ ___ ___ ___ -14 1 2 3 4 5 6 7 8 9 10 11 12

- 2 ___ ___ ___ ___ ___ -14 1 2 3 4 5 6 7

an = a1 + (n −1)d -14 = -2+(7-1)d -14 = -2+6d -12 = 6d d = -2

8 6 4 2 0 - 2 -4 -6 -8 -10 -12 -14 1 2 3 4 5 6 7 8 9 10 11 12

Answer: d = -2 and a1 = 8

Example 3: The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1

and d. We are going to look at another way to do the same problem.

a6 = a1 + (6 − 1)da12 = a1 + (12 − 1)d

−2 = a1 + 5d−14 = a1 + 11d

12 = -6d d = -2

Algebra II Unit 8 6

8 6 4 2 0 - 2 -4 -6 -8 -10 -12 -14 1 2 3 4 5 6 7 8 9 10 11 12

Answer: d = -2 and a1 = 8

Problems:1. The fifth term of an arithmetic sequence is -8 and the twelfth term is -22. Find a1 and d.

2. The sixth term of an arithmetic sequence is 12 and the eleventh term is -3. Find a1 and d.

3. The fourth term of an arithmetic sequence is 13 and the sixth term is 7. Find the first term and the common difference.

Algebra II Unit 8 7

Arithmetic Means

The terms between any two given terms of an arithmetic sequence are called arithmetic means. For example, in the sequence 2, 8, 14, 20, 26. ... the terms 8, 14, and 20 are the three arithmetic means between 2 and 26.

Example 1: Find three arithmetic means between 6 and 12.

6, ____, ____, ____, 12 1 2 3 4 5

an = a1 + (n −1)d

a5 = a1 + (5 −1)d

12 = 6 + 4d

6 = 4d

d =

64

= 32

Arithmetic Means: 6+ 32

=

€

152

€

152

+32

=182

= 9

€

9+32

=212

1. Find two arithmetic means between 3 and 15.

2. Find three arithmetic means between 3 and 13.

3. Find three arithmetic means between 6 and 11.

Algebra II Unit 8 8

Geometric Sequences

A geometric sequence is a sequence in which the ratio, r, between successive terms is always the same number.

Formula: an = a1 • rn−1

A geometric sequence: 4, 8, 16, 32, ... common ratio is 2.

Example 1: Find the eighth term of 4, 8, 16, 32, ...

an = a1 • rn−1

a8 = 4 • 28−1

a8 = 4 • 27

a8 = 4 •128a8 = 512

1. Find the common ratio for each geometric sequence.

A. 3, 9, 27

B. 8, 2, 12

, 18

2. Find the missing term in each geometric sequence:

A. 2, 8, 32, ____, ...

B. ____, 3, 1, 13

, ...

Algebra II Unit 8 9

3. The first term of a geometric sequence is 2. The common ratio is -2. Write the first four terms.

4. The first term of a geometric sequence is 14

. The common ratio is 2. Write the first four

terms.

5. Find the 8th term of the geometric sequence: 1, 2, 4, ...

6. Find the 6th term of the geometric sequence: -6, -3, −

32

, ...

7. A tank contains 8000 liters of water. Each day, one half of the remaining water is removed. How much water will be in the tank on the eighth day?

8. Jane owns 20 shares of Contex, Inc. If the stock splits (she receives 2 shares for every share she owns) every December for 5 years, how many shares will she own at the end of the fifth year? (At the end of the first year she owns 40 shares.)

9. Find the 5th term of the geometric sequence: 23

, 12

, 38

,...

10. Find the ninth term of the geometric sequence: 1, 3, 9, 27, ...

Algebra II Unit 8 10

Introduction to Series and Sigma NotationIf you add the terms of a sequence, the result is called a series:

Sequence: 3, 5, 7, 9, 11,...

Series: 3+5+7+9+11+ ...

Sn= the nth partial sum of a series

S1= 3 S2= 3+5 = 8 S3= 3+5+7 = 15 S4 = 3+5+7+9 = 24, and so forth.

Mathematicians invented a symbol telling us how to construct a partial sum:

Sigma: ∑

Let's see how to use sigma notation:

(2)(3k)

k=1

4∑ = (2)(31) + (2)(32) + (2)(33) + (2)(34)

= 6 + 18 + 54 + 162

= 240

Evaluate the expression by writing the terms and adding them up:

1.

(4k − 3)k=1

4∑ 2.

(3k − 2)

k=1

5∑

3.

(k + 6)k=1

3∑ 4.

5n

n=1

4∑


Sum of an Arithmetic Series

Example 1: Find the sum of the first fifteen terms of 2+5+8+...

a1 = 2n = 15d = 3

Now we need to find the fifteenth term:

an = a1 + (n −1)da15 = 2 + (15 − 1)(3)a15 = 2 + (14)(3)a15 = 2 + 42a15 = 44

Using our new formula:

sn =

n(a1 + an)2

s15 =

15(2 + 44)2

S15 =

15(46)2

S15 =

6902

S15 = 345


Example 2:

(7 − 2k)k=1

10∑

n=10 a1 = 7 − 2(1) = 7 − 2 = 5 a10 = 7 − 2(10) = 7 − 20 = −13

Sn =

n(a1 + an)2

S10 =

10(5 + (−13))2

S10 =

10(−8)2

S10 =

−802

= −40

Sn =

n(a1 + an)2

an = a1 + (n −1)d

1. Find the sum of the first sixteen terms of the arithmetic series: -7-5-3-...

2. Find the sum of the first ten terms of the arithmetic series: 7+12+17...

3. Find the sum of the first twenty terms of the arithmetic series: -13-11-9-...


4.

(3k +5)k=1

100∑

5.

−nn=1

200∑

6. In a lecture hall, the front row has 18 seats. Each succeeding row has 4 more seats than the row ahead of it. How many seats are there in the first 12 rows?

Sum of a Geometric Series

Example 1: Find the sum of the series 2+6+18+54+162.

n=5 a1 = 2 r = 3

sn =

a1(1− rn)1− r

s5 =

2(1−35)1−3

S5 =

2(−242)−2

S5 = 242


Example 2:

2kk=1

4∑

a1 = 2 n=4 r=2

Sn =

a1(1− rn)1− r

S5 =

2(1−24)1−2

S4 =

2(−15)−1

S4 = 30

Sn =

a1(1− rn)1− r

1. Find the sum of the first five terms of the geometric series: 4+8+16+...

2. Find the sum of the first five terms of the geometric series: 2-8+32-...

3. Find the sum of the geometric series: 1+4+16+64+256.

4.

3k−1k=1

4∑


5.

2kn=1

11∑

6. How much would you have saved at the end of seven days if you had set aside $.64 on the first day, $.96 on the second day, $1.44 on the third day and so on?

The Sum of an Infinite Geometric Series

This formula can be used to find the sum of an infinite geometric series when |r|<1.

Example 1: Find the sum of 5 −

53+

59−

527

+ ...

a1 = 5 r = − 1

3

s = a1

1− r

S =5

1+ 13

S=

543

€

S =51

•34

=154

= 3 34


Find the sum of each infinite geometric series:

1. 4+2+1+...

2. 12 + 4 +

43+ ...

3. 5 +

53+

59+ ...

Example: Drop a ball from 1 foot in the air. Suppose it consistently bounces up exactly 12

way

up the distance it fell. How far does the ball travel after the first bounce?

1

1st bounce+

++ ...

Startafter the first bounce

s∞ =

12+

12+

14+

14+

18+

18+

116

+1

16+ ...

s∞ = 2 1

2

+2 1

4

+ 2 1

8

+ 2 1

16

+ ...

s∞ = 1+ 1

2+

14+

18+ ...

s = a1

1− r a1 = 1

r = 1

2


S =1

1− 12

S =112

S =

11•

21= 2

1. A ball is dropped from a height of 12 meters. Suppose it consistently bounces up exactly 13

of the way up the distance it fell. How far does the ball travel after the first bounce?

2. A ball is thrown vertically upward a distance of 54 meters. After hitting the ground, it

rebounds 23

the distance fallen and continues to rebound in the same manner. What distance

does the ball cover before coming to rest?


Formulas

1. the nth term of an Arithmetic Sequence

2. The nth term of an Geometric Sequence an = a1 • rn−1

3. The Sum of the first n terms of an Arithmetic Series

Sn =n a1 + an( )

2

4. The Sum of the first n terms of a Geometric Series

Sn =a1 1− rn

1− r

5. The Sum of an infinite Geometric Series where -1<r<1

S =a1

1− r


Additional Problems: Please do your work on a separate piece of paper.

1. Find the first three terms of each sequence:A.

€

an = 6nB.

€

an = n3 + 2C.

€

an = n +1

2. Find the next 3 terms of the sequence.Find the rule for the nth term.

A. 4, 8, 12, 16, ...B. –20, –26, –32 , –38, ...

3. For each of the following sequences, determine if it is arithmetic, geometric, or neither. If it is arithmetic, find d. If it is geometric, find r.

A. 4, 7, 10, 13, .......

B. 3, 6, 12, 24, 48, ........

C. 2, 6, 24, 120, ......

D. 5, 53

, 59

, 527

,....

E. 1, 4, 9, 16, .........

4. Calculate

€

a100 for the sequence: 17, 22, 27, 32, ...

5. Calculate

€

a100 for the geometric sequence with first term a1 = 35 and common ratio r = 1.05.

6. The number 68 is a term in the arithmetic sequence with a1 = 5 and d = 3. Which term is it?

7. A geometric sequence has a1 = 17 and r = 2. If an = 34816, find n.


8. Find four arithmetic means between 55 and 85.

9. Find five arithmetic means between -91 and -67.

10. Find three arithmetic means between -257 and -397.

11.

(2n + 3)n=1

3∑ =

12.

n2

n=1

4∑

13. Find the 127th partial sum of the arithmetic series with a1 = 17 and d = 4.

14. Find S34 for the geometric series with a1 = 7 and r = 1.03.

15. Given the sequence: 15, 9, 3, -3, ... Find the 50th term.

16. Given the sequence: 4, 6, 9, ... Find the 25th term.

17. The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1 and d.

18. The first term of an arithmetic sequence is 3 and the common difference is 2. Find the 900th term.

19. Find the sum of the first 30 terms of a geometric series whose first term is 1 and whose common ratio is 2.

20. Find the 14th term of the geometric sequence: 1, 2, 4, ...

21.

€

(k + 2)k=1

3

∑ =

22. Find the sum of the first 15 terms: 2+5+8+...

23. Find the sum of the first sixteen terms of the series: -7-5-3-...

24. Find the sum of the first ten terms of the series: 7+12+17+22+...


25. Find the sum of the first twenty terms of the series: -13-11-9-7-...........

26.

5nn=1

4∑

27. Find the sum of the first 12 terms of the series: 1, 3, 9, 27...

28. Evaluate:

(3k +5)k=1

100∑

29. Evaluate:

−nn=1

200∑


algebra ii unit 8 1 - lsrhs.net 8.pdf · algebra ii unit 8 1. use the rule to write the first three...

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