Question

the academy for academic excellence has just announced their winner for the most successful student in mathematics. this student will receive a grant for their college education as well as national recognition for academic excellence in the field of mathematics. the grant will cover $12,177.17 of the cost of school per year for 5 years. if the initial amount is invested into an account over the five years, paying an annual interest rate of 5.5%, and you are to withdraw from this account until the remaining balance after the 5 years is zero dollars, determine the initial amount of the grant. round your answer to the nearest dollar.

Explanation:

Step1: Identify the formula

The formula for the present value of an ordinary annuity is \(PV = PMT\times\frac{1-(1 + r)^{-n}}{r}\), where \(PV\) is the present value (initial amount of the grant), \(PMT\) is the annual payment (\(PMT=\$12177.17\)), \(r\) is the annual interest rate (\(r = 0.055\)), and \(n\) is the number of years (\(n = 5\)).

Step2: Substitute the values into the formula

$$
LATEXBLOCK0
$$

First, calculate \((1 + 0.055)^{-5}=(1.055)^{-5}\approx0.765164\)
Then, \(1-(1.055)^{-5}=1 - 0.765164 = 0.234836\)
Next, \(\frac{1-(1.055)^{-5}}{0.055}=\frac{0.234836}{0.055}\approx4.269745\)
Finally, \(PV=12177.17\times4.269745\approx52000\)

Answer:

\(\$52000\)