Suppose a young couple deposits $900 at the end of each quarter in an account that earns 7.2%, compounded quarterly, for a period of 9 years. After the 9 years, they start a family and find they can contribute only $200 per quarter. If they leave the money from the first 9 years in the account and continue to contribute $200 at the end of each quarter for the next
18 1/2 years, how much will they have in the account (to help with their child's college expenses)? (Round your answer to the nearest cent.)

You want to know the future value of an ordinary annuity earning 7.2% if quarterly payments are $900 for the first 9 years and $200 for the next 18 1/2 years.

Future value

To answer this question, we need two future value formulas:

A = P(1 +r)^n . . . . . . . . . future value of an investment of P
A = P((1 +r)^n -1)/r . . . . . future value of a series of payments of P

In each formula, r is the periodic interest rate, and n is the number of periods.

Application

For this problem, each period is one quarter, so the quarterly interest rate is 7.2%/4 = 1.8%.

For the first 9 years, the number of quarters is 4·9 = 36. For the last 18 1/2 years, the number of quarters is 4·(18 1/2) = 74.

At the end of 9 years, the value of the first series of payments is ...

A = 900(1.018^36 -1)/0.018 ≈ 45036.4078

The value of the annuity after 18 1/2 more years is the sum of the future values of this amount and the series of reduced payments.

A = 45036.4078(1.018^74) +200(1.018^74 -1)/0.018 = 199105.07

The account value will be about $199,105.07.