How much would you need to deposit in an account now in order to have \( \$ 4000 \) in the account in 15 years? Assume the account earns \( 2 \% \) interest compounded monthly.

To find out how much you need to deposit now in order to have \( \$4000 \) in 15 years with an interest rate of \( 2\% \) compounded monthly, we can use the formula for the present value of a future sum: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) is the present value (the amount to deposit now), - \( FV \) is the future value (the amount desired in the future, which is \( \$4000 \)), - \( r \) is the annual interest rate (as a decimal, so \( 0.02 \) for \( 2\% \)), - \( n \) is the number of times the interest is compounded per year (monthly compounding means \( n = 12 \)), - \( t \) is the number of years the money is invested (which is \( 15 \) years). Now, substituting the values into the formula: \[ PV = \frac{4000}{(1 + 0.02/12)^{12 \times 15}} \] Calculating \( r/n \): \[ r/n = \frac{0.02}{12} = 0.00166667 \] Calculating \( nt \): \[ nt = 12 \times 15 = 180 \] Now we can calculate \( (1 + r/n)^{nt} \): \[ (1 + 0.00166667)^{180} \] Calculating \( 1 + 0.00166667 \): \[ 1 + 0.00166667 = 1.00166667 \] Now raising it to the power of \( 180 \): \[ (1.00166667)^{180} \approx 1.348850 \] Now substituting this back into the present value formula: \[ PV = \frac{4000}{1.348850} \approx 2962.36 \] Thus, the amount you need to deposit now is approximately: \[ \boxed{2962.36} \]