GeometricSequences and

Series

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

2. Geometric Sequences and Series

a1 a2 a3 a4 a5 a6anan - 1

+ d + d + d + d + d

r

+ d

r r r r r

Geometric Sequences (Type 2)• In geometric sequences, you multiply by a common

ratio (r) each time.

•1, 2, 4, 8, 16, ... multiply by 2•27, 9, 3, 1, 1/3, ...Divide by 3 which means multiply by 1/3

ie ru

u

n

n 1

The nth term of an geometric sequence is denoted by the formula

1 nn aru

Where a is the 1st term and r is the common ratio

The sum of the first n terms of a geometric series is found by using:

)()(

r

raS

n

n

1

1

Note if r>1 then we can use the formula

Which is more convenient )1(

)1(

r

raS

n

n

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1n 1

n

n 1

nth term of geometric sequence

sum of the first n terms of geometric seq

a a r

1 rue ce S a

1n

r

1 n

n

or

sum of the first n terms of geometric sequenca ra

S1

er

Find the next three terms of 2, 3, 9/2, ___, ___, ___

3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3

1.5 geometric r2 3 2

3 3 3 3 3 3

2 2 2

92, 3, , , ,

2

9 9 9

2 2 2 2 2 2

92, 3, , ,

27 81 243

4 8,

2 16

1 9

1 2If a , r , find a .

2 3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

x

9

NA

2/3

n 1n 1a a r

9 1

9

1 2a

2 3

8

9 8

2a

2 3

7

8

2

3

128

6561

Example 2

Find u10 for the geometric sequence 144, 108, 81, 60¾, …

Answer

a = 144 and r = 4

3

12

9

144

108

u10 = arn-1

= 144 (¾)9 = 10.812…

Example 3

Find S19 for the geometric sequence 3-6+12-24+…

Answer

a = 3 and r = 23

6

5242893

5242893

3

52428813

21

213

1

1 19

)())((

)())((

)()(

r

raS

n

n

2 4 1

2Find a a if a 3 and r

3

-3, ____, ____, ____

2Since r ...

3

4 83, 2, ,

3 9

2 4

8 10a a 2

9 9

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

9

1

a 2 2

8

2 2

16 2

5 2 4If a 32 2 and r 2, find a to 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

1

32 2 a 2

4

132 2 a 2

132 2 4a

1a 8 2

2a 8 2( 2) 16

23a 8 2( 2) 16 2

34a 8 2( 2) 32

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

n

n 1

1 rS a

1 r

7

7

11

1s

2 1

212

71

1 22

1

12

7

11

2

63

64

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

3, 7, 11, …, 51 Finite Arithmetic

n 1 n

1

nS a a

2n 1

n a d2

1, 2, 4, …, 64 Finite Geometric

n1 n

n 1

a ra1 rS a

1 r 1 r

1, 2, 4, 8, … Infinite Geometricr 1 or r -1

No Sum

1 1 13,1, , , ...

3 9 27Infinite Geometric

-1 < r < 11a

S1 r

13.5 Infinite Geometric Series

|r| 1

|r| < 1

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 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

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

0S 2 5 500 50

504

15

450

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

An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest?

25

20

16

25

20

16

S

254

15

2 250