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Geometric Sequences and Series

Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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Page 1: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

GeometricSequences and

Series

Page 2: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 3: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 4: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 5: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 6: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 7: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 8: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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…

Page 9: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 10: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 11: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 12: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 13: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 14: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 15: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 16: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

8 16 2r 2 2

82 2 r 1 No

NO SUM

Page 17: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 18: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

Find the sum, if possible: 2 4 8

...7 7 7

4 87 7r 22 47 7

r 1 No

NO SUM

Page 19: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

Find the sum, if possible: 5

10 5 ...2

55 12r

10 5 2 1 r 1 Yes

1a 10S 20

11 r 12

Page 20: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 21: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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

Page 22: Geometric Sequences and Series. Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms

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