In this situation, you are making regular monthly contributions to an account (an IRA) that earns interest compounded monthly. The correct formula to use for the future value of a series of regular deposits (an annuity) is:
\[ A = P \frac{(1 + r)^{nt} - 1}{r} \]
Where:
- \( A \) is the future value of the investment/loan, including interest,
- \( P \) is the amount of money deposited each period (in this case, $60),
- \( r \) is the monthly interest rate (annual rate divided by the number of periods per year),
- \( n \) is the number of compounding periods per year,
- \( t \) is the number of years the money is invested or borrowed.
For your specific situation:
- The annual interest rate is 6%, so the monthly interest rate \( r = \frac{0.06}{12} = 0.005 \).
- You will be making deposits for \( 40 \) years (from age 24 to age 64), so \( t = 40 \) and \( n = 12 \).
Based on this, the formula simplifies to:
\[ A = 60 \frac{(1 + 0.005)^{12 \times 40} - 1}{0.005} \]
Thus, the formula to determine the future value of the account using regular monthly deposits and compound interest is:
\[ A = P \frac{(1 + r)^{nt} - 1}{r} \]
So, among the options provided, you would select \( A = \frac{(P[(1+r)^{t}-1])}{r} \) with the understanding that \( r \) represents the monthly interest rate and \( t \) should actually account for the number of periods (i.e., \( nt \)).
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