Copyright © 2011 Pearson Education, Inc. Slide 11.3-1

• A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number, called the common ratio.

• The series of wages 1, 2, 4, 8, 16 … is an example of a geometric sequence in which the first term is 1 and the common ratio is 2.

11.3 Geometric Sequences and Series

Copyright © 2011 Pearson Education, Inc. Slide 11.3-2

11.3 Finding the Common Ratio

• In a geometric sequence, the common ratio can be found by choosing any term except the first and dividing by the preceding term.

The geometric sequence 2, 8, 32, 128, …

has common ratio r = 4 since

8 32 128... 4

2 8 32

Copyright © 2011 Pearson Education, Inc. Slide 11.3-3

11.3 Geometric Sequences and Series

nth Term of a Geometric Sequence

In a geometric sequence with first term a1 and common ratio r, neither of which is zero, the nth term is given by

11 .n

na a r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-4

11.3 Using the Formula for the nth Term

Example Find a5 and an for the geometric

sequence 4, –12, 36, –108 , …

Solution Here a1= 4 and r = 36/ –12 = – 3. Using

n=5 in the formula

In general

5 1 45 4 ( 3) 4 ( 3) 324a

1 11 4 ( 3)n n

na a r

11

nna a r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-5

Your turn.

Find a5 and an.

1.

2.

3.

€

a1 = 20,a2 =10,a3 = 5.

€

a1 = 250,a2 = −25,a3 = −2.5.

€

a1 =1,a2 = −4,a3 =16.

Copyright © 2011 Pearson Education, Inc. Slide 11.3-6

11.3 Modeling a Population of Fruit Flies

Example A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. Suppose there were 100 insects in the first generation. How many would there be in the fourthgeneration?

Solution The populations form a geometric sequence

with a1= 100 and r = 1.5 . Use n = 4 in the formula

for an..

In the fourth generation, the population is about 338 insects.

3 34 1 100(1.5) 337.5a a r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-7

11.3 Geometric Sequences and Series

Sum of the First n Terms of an Geometric Sequence

If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by

where 1(1 )

1

n

n

a rS

r

1.r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-8

11.3 Geometric Series

• A geometric series is the sum of the terms of a geometric sequence .

In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:

1 2 3 4

2 3100 100(1.5) 100(1.5) 100(1.5)

813

a a a a

Copyright © 2011 Pearson Education, Inc. Slide 11.3-9

11.3 Finding the Sum of the First n Terms

Example Find

Solution This is the sum of the first six terms of a

geometric series with and r = 3.

From the formula for Sn ,

.

11 2 3 6a

6

1

2 3i

i

6

6

6(1 3 ) 6(1 729) 6( 728)2184

1 3 2 2S

Copyright © 2011 Pearson Education, Inc. Slide 11.3-10

Practice

Find the sum of the finite geometric series.

1.

2.

3. €

2 • 4 i

i=1

10

∑

€

6 •1

2

⎛

⎝ ⎜

⎞

⎠ ⎟i

i=1

8

∑

€

1

5• −

5

4

⎛

⎝ ⎜

⎞

⎠ ⎟i

i=1

12

∑

Copyright © 2011 Pearson Education, Inc. Slide 11.3-11

11.3 An Infinite Geometric Series

Given the infinite geometric sequence

the sequence of sums is S1 = 2, S2 = 3, S3 = 3.5, …

1 1 1 12, 1, , , , ,...

2 4 8 16

The calculator screen shows more sums, approaching a value of 4. So

1 12 1 ... 4

2 4

Copyright © 2011 Pearson Education, Inc. Slide 11.3-12

11.3 Infinite Geometric Series

Sum of the Terms of an Infinite Geometric Sequence

The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where –1 < r < 1, is given by

.1

1

aS

r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-13

11.3 Finding Sums of the Terms of Infinite Geometric Sequences

Example Find

Solution Here and so

.

1

3

5

i

i

1

3

5a

1

1

33 35

35 1 215

i

i

a

r

3

5r

Copyright © 2011 Pearson Education, Inc. Slide 11.3-14

Infinite Geometric Series Practice

Find the sum of the series.

1.

2. €

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟i

i=1

∞

∑

€

2

3•

1

10

⎛

⎝ ⎜

⎞

⎠ ⎟i

i=1

∞

∑

Copyright © 2011 Pearson Education, Inc. Slide 11.3-15

11.3 Annuities

Future Value of an Annuity

The formula for the future value of an annuity is

where S is the future value, R is the payment at the end of each period, i is the interest rate in decimal form per period, and n is the number of periods.

1 1,

ni

S Ri

Copyright © 2011 Pearson Education, Inc. Slide 11.3-16

The Value of an Annuity

You have an annuity with a monthly payment of $250 that pays a annual interest rate of 6%. How much will it be worth after 10 years? i = .5% = .005 n = 120

€

S120 = 2501+ .005( )

120−1

.005

⎡

⎣ ⎢

⎤

⎦ ⎥