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Q 88.

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Found in: Page 810

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# Education IRA: On January 1, 1999, John’s parents decided to place $45 at the end of each month into an Education IRA.(a) Find a recursive formula that represents the balance at the end of each month if the rate of return is assumed to be 6% per annum compounded monthly.(b) How long will it be before the value of the account exceeds$4000?(c) What will be the value of the account in 16 years when John goes to college?

(a) The recursive formula for interest compounded annually is:

${p}_{n}=1.005{p}_{n-1}+45\phantom{\rule{0ex}{0ex}}&{p}_{0}=0$

(b) The value of account exceeds $4,000 after 75 months. (c) The value of account in 192 months will be$14,216.36.

See the step by step solution

## Step 1. Write the given information.

After the end of every month, John's parents' deposit $45 i.e., A=$45.

The interest rate compounded quarterly, r = 6.

## Step 2. Use the formula for compound interest to compute the recursive formula.

Using the compound interest formula as:

${p}_{n}=\left(1+\frac{r}{n}\right){p}_{n-1}+A\phantom{\rule{0ex}{0ex}}⇒{p}_{n}=\left(1+\frac{0.06}{12}\right){p}_{n-1}+45\phantom{\rule{0ex}{0ex}}⇒{p}_{n}=1.005{p}_{n-1}+45\phantom{\rule{0ex}{0ex}}&{p}_{0}=0$

## Step 4. Calculate the number of months in 16 years and then calculate the amount in the account.

Number of months in 16 years:

$12×16=192$

The amount in the account after 192 months will be:

${p}_{192}=1.005×{p}_{191}+45\phantom{\rule{0ex}{0ex}}⇒{p}_{192}=14,216.36$