Suppose you could make a single "lump sum" deposit of $\$ 5172$, in an investment that provides an Annual Percentage Rate(APR) of $3 \%$ compounded quarterly. Determine the Future Value(FV) of the investment after 21 years.
Solution
Step 1 ：Given that the principal amount \(P = 5172\), the annual interest rate \(r = 0.03\), the number of times that interest is compounded per year \(n = 4\), and the time the money is invested for in years \(t = 21\).
Step 2 ：The formula for the future value (FV) of an investment compounded quarterly is given by \(FV = P * (1 + r/n)^{nt}\).
Step 3 ：Substitute the given values into the formula: \(FV = 5172 * (1 + 0.03/4)^{4*21}\).
Step 4 ：Calculate the value inside the parentheses: \(1 + 0.03/4 = 1 + 0.0075 = 1.0075\).
Step 5 ：Calculate the exponent: \(4*21 = 84\).
Step 6 ：So, the formula becomes: \(FV = 5172 * (1.0075)^{84}\).
Step 7 ：Calculate the power: \((1.0075)^{84} ≈ 1.9994\).
Step 8 ：Finally, multiply this by the principal amount: \(FV = 5172 * 1.9994 ≈ 10340.68\).
Step 9 ：So, the future value of the investment after 21 years would be approximately \(\boxed{10340.68}\).
