Suppose you want to have \( \$ 600,000 \) for retirement in 35 years. Your account earns \( 5 \% \) interest. How much would you need to deposit in the account each month?

We use the future value of an annuity formula given by \[ FV = PMT \cdot \frac{(1+r)^n - 1}{r}, \] where - \(FV\) is the future value (in this case, \(\$600,\!000\)), - \(PMT\) is the monthly deposit, - \(r\) is the monthly interest rate, - \(n\) is the total number of deposits. **Step 1. Determine the parameters** - The annual interest rate is \(5\%\), so the monthly interest rate is \[ r = \frac{0.05}{12} \approx 0.00416667. \] - The total number of months in 35 years is \[ n = 35 \times 12 = 420. \] **Step 2. Solve for \(PMT\)** Rearrange the future value formula to solve for \(PMT\): \[ PMT = \frac{FV \cdot r}{(1+r)^n - 1}. \] Substitute the known values: \[ PMT = \frac{600,\!000 \cdot 0.00416667}{(1+0.00416667)^{420} - 1}. \] **Step 3. Compute \((1+r)^n\)** Calculate \[ (1+0.00416667)^{420}. \] Taking natural logarithms, \[ \ln(1.00416667) \approx 0.004158, \] so \[ (1.00416667)^{420} = e^{420 \times 0.004158} \approx e^{1.74636} \approx 5.7375. \] Then, \[ (1+r)^n - 1 \approx 5.7375 - 1 = 4.7375. \] **Step 4. Substitute and calculate \(PMT\)** Now, the numerator is \[ 600,\!000 \times 0.00416667 = 2500. \] Thus \[ PMT = \frac{2500}{4.7375} \approx 527.99. \] **Final Answer:** You would need to deposit approximately \(\$528\) each month.