Geometric Sequences 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

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

• Example: 3, 6, 12, 24, 48, ...

• The terms have a common ratio of 2.

• The common ratio is the number r.

• To find the common ratio you use an+1 ÷ an

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

Find the next two terms of 2, 6, 18, ___, ___6 – 2 vs. 18 – 6… not arithmetic

2, 6, 18, 54, 162

Find the next two terms of 80, 40, 20, ___, ___40 – 80 vs. 20 – 40… not arithmetic

80, 40, 20, 10, 5

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

92, 3, , ,

27 81 243

4 8,

2 16

Find the next two terms of -15, 30, -60, ___, ___30 – -15 vs. -60 – 30… not arithmetic

-15, 30, -60, 120, -240

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

-3

an

8

NA

-2

n 1n 1a a r

Find the 8th term if a1 = -3 and r = -2.

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

??

an

10

NA

3n 1

n 1a a r

Find the 10th term if a4 = 108 and r = 3.

4

Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, …

3

41a First term

r common ratio

n 1n 1a a r

Geometric Mean: The terms between any two nonconsecutive terms of a geometric sequence.

Ex. 2, 6, 18, 54, 162

6, 18, 54 are the Geometric Mean between 2 and 162

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

r

n 1n 1a a r

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

forms a geometric sequence

Geometric Series: An indicated sum of terms in a geometric sequence.

Example:

Geometric Sequence

3, 6, 12, 24, 48

VS Geometric Series

3 + 6 + 12 + 24 + 48

RecallVocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

Application: Suppose you e-mail a joke to three friends on Monday. Each of those friends sends the joke to three of their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday, and so on.

Monday

Tuesday

# New people that receive joke Day of Week Total # of people that received joke

  Monday  

  Tuesday  

  Wednesday  

3 3

9 3 + 9 = 12

27 12 + 27 = 39

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

3

10

Sn

NA

Find the sum of the first 10 terms of the geometric series 3 - 6 + 12 – 24+ …

-2

In the book Roots, author Alex Haley traced his family history back many generations to the time one of his ancestors was brought to America from Africa. If you could trace your family back 15 generations, starting with your parents, how many ancestors would there be?

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

2

15

Sn

NA

2

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

a1

8

39,360

NA

3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

15,625

??

Sn

-5

Recall the properties of exponents. When multiplying like bases add exponents

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

15,625

??

Sn

-5

B

nn A

a

UPPER LIMIT(NUMBER)

LOWER LIMIT(NUMBER)

SIGMA(SUM OF TERMS) NTH TERM

(SEQUENCE)

INDEX

n

n 0

4

0.5 2

00.5 2 10.5 2 20.5 2 30.5 2 40.5 2 33.5

If the sequence is geometric (has a common ratio) you can use the Sn formula

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

5+20 = 6

5

Sn

2

5+25 = 37

Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1

Geometric, r = ½ n 1

n 1a a r n 1

n

1a 16

2

n 1

n

5

1

116

2

Infinite Series

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