Ann and Tom want to establish a fund for their grandson's college education. What lump sum must they deposit at a \( 12 \% \) annual interest rate, compounded annually, in order to ha \( \$ 20,000 \) in the fund at the end of 15 years? They should deposit \( \$ \square \) (Round up to the nearest cent.)

To determine the lump sum that Ann and Tom need to deposit, we can use the formula for future value in compound interest: \[ FV = PV \times (1 + r)^n \] Where: - \( FV \) is the future value (\$20,000), - \( PV \) is the present value (the lump sum they need to deposit), - \( r \) is the annual interest rate (12% or 0.12), - \( n \) is the number of years the money is invested (15). Rearranging the formula to find the present value gives us: \[ PV = \frac{FV}{(1 + r)^n} \] Plugging in the values: \[ PV = \frac{20000}{(1 + 0.12)^{15}} \] First, calculate \( (1 + 0.12)^{15} \): \[ (1.12)^{15} \approx 5.4739 \] Now, substitute back into the equation: \[ PV = \frac{20000}{5.4739} \approx 3642.29 \] Thus, Ann and Tom should deposit approximately \( \$ 3,642.29 \) to reach their goal of \( \$ 20,000 \) in 15 years.