12.3 Geometric Sequences and Series

©2001 by R. Villar

All Rights Reserved

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

Geometric Sequences

MULTIPLYTo get next term

Arithmetic Series

Sum of Terms

35

12

27.2

3 9

Geometric Series

Sum of Terms

62

20 / 3

85 / 64

9.75

Geometric Sequences and Series

Geometric Sequence: sequence whose consecutive terms have a common ratio.

1, 3, 9, 27, 81, 243, ...

The terms have a common ratio of 3.

The common ratio is the number r = .

Example Is the sequence geometric? 4, 6, 9, 13.5, 20.25, 30.375…

Yes, the common ratio is 1.5To find any term in a geometric sequence, use the

formula an = a1 rn–1 where r is the common ratio.

n

n

a

a 1

Example: Find the common ration of a geometric sequence if the first term is 8 and the 11th term is 1/128.

an = a1 rn–1 a1 = 9 r = 1.2 a9 = 9 • 1.211

a12 = 66.87

Example: Find the twelfth term of the geometric sequence whose first term is 9 and whose common ratio is 1.2.

an = a1 rn–1 a1 = 8 a11 = 1/128 1/128 = 8 r 10

1/1024 = r10

½ = r

5 2If a 32 2 and r 2, find a

____, , ____,________ ,32 2

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

5

NA

32 2

2n 1

n 1a a r

5 1

32 2 x 2

4

32 2 x 2

32 2 x4

8 2 x

9Find a of 2, 2, 2 2,...

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

9

NA

2

2 2 2r 2

22

n 1n 1a a r

9 1

x 2 2

8

x 2 2

x 16 2

Find two geometric means between –2 and 54

-2, ____, ____, 54

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

-2

54

4

NA

x

n 1n 1a a r

1454 2 x

327 x 3 x

The two geometric means are 6 and -18, since –2, 6, -18, 54

forms an geometric sequence

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1

n 1

n1

n

nth term of geometric sequence

sum of n terms of geometric sequ

a a r

a r 1S

r 1ence

Which can be simplified to:

1

1

in

iira

To find the sum of a geometric series, we can use summation notation.

a1

1 r n

1 r

Example: Evaluate the sum of:

Convert this to

30(i1

6

0.2)i

a1

1 r n

1 r

ai

i1

n

ri 1

61 0.26

1 0.2

61 .000064

0.8

= 7.49952

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

7

x

NA

11184r

1 1 22 4

n1

n

a r 1S

r 1

71 12 2

x12

1

1

71 12 2

12

1

63

64

1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum

3, 7, 11, …, 51 Finite Arithmetic n 1 n

nS a a

2

1, 2, 4, …, 64 Finite Geometric n

1

n

a r 1S

r 1

1, 2, 4, 8, … Infinite Geometricr > 1r < -1

No Sum

1 1 13,1, , , ...

3 9 27Infinite Geometric

-1 < r < 11a

S1 r

Find the sum, if possible: 1 1 1

1 ...2 4 8

1 112 4r

11 22

1 r 1 Yes

1a 1S 2

11 r 12

Find the sum, if possible: 2 2 8 16 2 ...

8 16 2r 2 2

82 2 1 r 1 No

NO SUM

Find the sum, if possible: 2 1 1 1

...3 3 6 12

1 113 6r

2 1 23 3

1 r 1 Yes

1

2a 43S

11 r 312

Find the sum, if possible: 2 4 8

...7 7 7

4 87 7r 22 47 7

1 r 1 No

NO SUM

Find the sum, if possible: 5

10 5 ...2

55 12r

10 5 2 1 r 1 Yes

1a 10S 20

11 r 12

The Bouncing Ball Problem – Version A

A ball is dropped from a height of 50 feet. It rebounds 4/5 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?50

40

32

32/5

40

32

32/5

40S 45

504

10

1554

The Bouncing Ball Problem – Version B

A ball is thrown 100 feet into the air. It rebounds 3/4 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?

100

75

225/4

100

75

225/4

10S 80

100

4 43

1

0

10

3