Intermediate Algebra (11th edition)

FIGURE 2shows how a graphing calculator can store the terms in a list and then

find the sum of these terms. The figure supports the result of Example 6.

OBJECTIVE 5 Apply the formula for the future value of an ordinary

annuity.A sequence of equal payments made over equal periods is called an annuity.

If the payments are made at the end of the period, and if the frequency of payments is

the same as the frequency of compounding, the annuity is called an ordinary annuity.

The time between payments is the payment period,and the time from the beginning

of the first payment period to the end of the last is called the term of the annuity.

The future value of the annuity,the final sum on deposit, is defined as the sum of

the compound amounts of all the payments, compounded to the end of the term.

We state the following formula without proof.

SECTION 12.3 Geometric Sequences 695

FIGURE 2

Future Value of an Ordinary Annuity

The future value of an ordinary annuity is

where Sis the future value,

Ris the payment at the end of each period,

i is the interest rate per period, and

nis the number of periods.

SRc

11 i 2 n 1

i

d,

Applying the Formula for the Future Value of an Annuity

(a)Igor Kalugin is an athlete who believes that his playing career will last 7 yr. He

deposits $22,000 at the end of each year for 7 yr in an account paying 6%

compounded annually. How much will he have on deposit after 7 yr?

Igor’s payments form an ordinary annuity with and

The future value of this annuity (from the formula) is

or $184,664.43. Use a calculator.

(b)Amy Loschak has decided to deposit $200 at the end of each month in an account

that pays interest of 4.8% compounded monthly for retirement in 20 yr. How

much will be in the account at that time?

Because the interest is compounded monthly, Also, and

The future value is

D = 80,335.01, or $80,335.01.

a 1 +

0.048

12

b

121202

- 1

0.048

12

S= 200 T

n= 121202.

i= 0.048 12. R= 200

= 184,664.43,

S= 22,000c

1 1.06 27 - 1

0.06

d

R= 22,000,n= 7, i=0.06.

NOW TRY EXAMPLE 7

EXERCISE 7

(a)Billy Harmon deposits
$600 at the end of each
year into an account
paying 2.5% per yr,
compounded annually.
Find the total amount on
deposit after 18 yr.

(b)How much will be in Billy
Harmon’s account after
18 yr if he deposits $100
at the end of each month at
3% interest compounded
monthly?

NOW TRY ANSWERS

7. (a)$13,431.81 (b)$28,594.03

NOW TRY

OBJECTIVE 6 Find the sum of an infinite number of terms of certain

geometricsequences.Consider an infinite geometric sequence such as

1

3

,

1

6

,

1

12

,

1

24

,

1

48

,Á.