Let's break down the problem step by step. **Given:** - Withdrawals: $4000 each on son's 18th, 19th, 20th, and 21st birthdays (i.e., at the end of years 18, 19, 20, and 21). - Fund earns 9% interest, compounded annually. - Equal deposits are made at the end of each year, from year 1 through year 17 (i.e., 17 deposits). - We are to find the amount of each deposit, \( R \), such that the fund can meet these withdrawals. --- ### **Step 1: Find the Present Value (at time 0) Needed for Withdrawals** **First, find the present value (PV) at time 17 of all the withdrawals:** These are 4 withdrawals of $4000 each, at the **end** of years 18, 19, 20, and 21. At time 17, the value needed is: \[ PV_{17} = 4000 \cdot (P/A,\,9\%,\,4) \] Where \((P/A,\,9\%,\,4)\) is the present value of an ordinary annuity of 4 payments at 9%. \[ (P/A,\,9\%,\,4) = \frac{1 - (1+0.09)^{-4}}{0.09} \] \[ (1+0.09)^{-4} = (1.09)^{-4} = \frac{1}{(1.09)^4} \approx \frac{1}{1.4116} \approx 0.70843 \] \[ (P/A) = \frac{1 - 0.70843}{0.09} = \frac{0.29157}{0.09} \approx 3.2397 \] \[ PV_{17} = 4000 \times 3.2397 = \$12,958.80 \] **This is the amount needed at the end of year 17 to fund all withdrawals.** --- ### **Step 2: Find the Series of Annual Deposits to Reach This Amount** The father makes 17 equal deposits at the end of each year (from year 1 to year 17). We need to find the size \( R \) of each deposit so that the amount accumulated at the end of year 17 is $12,958.80. The future value of an ordinary annuity (deposits) at the end of year 17: \[ FV = R \cdot \frac{(1+i)^n - 1}{i} \] Set this equal to $12,958.80, with \( n = 17 \), \( i = 0.09 \): \[ R \cdot \frac{(1.09)^{17} - 1}{0.09} = 12,958.80 \] Calculate \((1.09)^{17}\): \[ \ln(1.09^{17}) = 17 \cdot \ln(1.09) \approx 17 \cdot 0.08618 = 1.4651 \] \[ (1.09)^{17} = e^{1.4651} \approx 4.3274 \] So, \[ \frac{(1.09)^{17} - 1}{0.09} = \frac{4.3274 - 1}{0.09} = \frac{3.3274}{0.09} = 36.971 \] Now, \[ R \cdot 36.971 = 12,958.80 \] \[ R = \frac{12,958.80}{36.971} \approx 350.49 \] --- ### **Final Answer** \[ \boxed{R = \$350.49} \] **The father should deposit \$350.49 at the end of each year for 17 years.** --- ### **Summary of Steps** 1. **Calculate the amount needed at the end of year 17:** Find the present value of the 4 withdrawals at year 17 (\$12,958.80). 2. **Set up a savings annuity:** Use the future value formula for an ordinary annuity to determine the annual deposit required to reach this amount in 17 years. 3. **Solve for the deposit:** \( R = \$350.49 \) per year (rounded to two decimals).