Chapter 10 Further Topics in Algebra

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SECTION 10.3

Geometric Sequences and Series

OBJECTIVES 1 2 3 4 Identify a geometric sequence and find its common ratio.

Find the sum of a finite geometric sequence.

Solve annuity problems.

Find the sum of an infinite geometric sequence.

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DEFINITION OF A GEOMETRIC SEQUENCE The sequence

a

1 ,

a

2 ,

a

3 ,

a

4 , … ,

a n , …

is a

geometric sequence

, or a

geometric progression,

r

if there is a number each term except the first in the sequence is obtained by multiplying the previous term by . The number

r

is called the

r

such that

common ratio

of the geometric sequence.

a n

 1 

r

,

a n n

 1

3

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RECURSIVE DEFINITION OF A GEOMETRIC SEQUENCE A geometric sequence formula

a a

1

n

,

a

+ 1 2 , =

a

3

ra

,

n a

4 , , … ,

n

≥ 1

a n , …

can be defined recursively. The recursive defines a geometric sequence with first term

a

1 and common ratio

r

.

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THE GENERAL TERM OF A GEOMETRIC SEQUENCE Every geometric sequence can be written in the form

a

1 ,

a

1

r

,

a

1

r

2 ,

a

1

r

3 ,

… , a

1

r n

− 1 , … where

r

is the common ratio. Since

a

1 =

a

1 (1) =

a

1

r

0 , the

n

th term of the geometric sequence

is

a n

=

a

1

r n

–1 , for

n

≥ 1.

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 3 Finding Terms in a Geometric Sequence For the geometric sequence 1, 3, 9, 27, …, find each of the following: a. a 1 b. r c. a

n

Solution

a.

The first term of the sequence is given

a

1 = 1.

b.

Find the ratio of any two consecutive terms:

c.

a n



a

1

r n

 1 

r

3

n

3  1     3.

3

n

 1 © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Finding a Particular Term in a Geometric Sequence Find the 23rd term of a geometric sequence whose first term is 10 and whose common ratio is 1.2.

Solution © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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SUM OF THE TERMS OF A FINITE GEOMETRIC SEQUENCE Let

a

1 ,

a

2 ,

a

3 ,

… a n

be the first

n

terms of a geometric sequence with first term

a

1 and common ration

r

. The sum

S n

is of these terms

S n



i n

  1

a r

1

i

 1 

a

1 1  1  

r r n

 ,

r

 1.

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EXAMPLE 5 Finding the Sum of Terms of a Finite Geometric Sequence Find each sum.

a.

i

15   1 Solution

 

i

 1

b.

i

15   1 a. a 1 = 5,

r S

15 

i

15   1 = 0.7,

n a r

1

i

 1 = 15  5    1 

 

15     16.588

b.

i

15   1 

i

15   1

 

i

 1   11.6116

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i

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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VALUE OF AN ANNUITY Let

P

represent the payment in dollars made at the end of each of

n

compounding periods per year, and let

i

be the annual interest rate. Then the value of

A

years is:

A



P

of the annuity after          1 

i n i n

  

nt

 1      

t

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EXAMPLE 6 Finding the Value of an Annuity An individual retirement account (IRA) is a common way to save money to provide funds after retirement. Suppose you make payments of $1200 into an IRA at the end of each year at an annual interest rate of 4.5% per year, compounded annually. What is the value of this annuity after 35 years?

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EXAMPLE 6 Finding the Value of an Annuity Solution

P

= $1200 ,

i A

= 0.045

 1200 and    1 

t

= 35 0.045

1 years      35  1 0.045

1  $97,795 .

94 The value of the IRA after 35 years is $97,795.94.

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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SUM OF THE TERMS OF AN INFINITE GEOMETRIC SEQUENCE If |

r

| < 1, the infinite sum

a

1 +

a

1

r

+

a

1

r

2 +

a

1

r

3 +

…

+

a

1

r n

–1 + … is given by

S

  

i

 1

a

1

r i

 1 

a

1 1 

r

.

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EXAMPLE 7 Finding the Sum of an Infinite Geometric Series Find the sum 2 2 9 8 27 32  .

Solution

a

1  2 and

r

 3 2 2  3 4 Since |

r

| < 1, use the formula:

S

 1

a

1 

r

 1  2 3 4  8 © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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