You want to be able to withdraw \( \$ 20,000 \) each year for 15 years. Your account earns \( 4 \% \) interest. How much do you need in your account at the beginning?

Given that you wish to withdraw \( \$20,\!000 \) each year for 15 years and your account earns \(4\%\) interest per year, we need to determine the present value (PV) of this annuity. The formula for the present value of an annuity is \[ \text{PV} = P \times \frac{1 - (1+r)^{-n}}{r} \] where \(P\) is the annual payment, \(r\) is the annual interest rate (in decimal form), and \(n\) is the number of years. **Step 1. Identify the variables:** - \(P = 20,\!000\) - \(r = 0.04\) - \(n = 15\) **Step 2. Plug the values into the formula:** \[ \text{PV} = 20,\!000 \times \frac{1 - (1+0.04)^{-15}}{0.04} \] **Step 3. Simplify the expression:** First, calculate \(1+0.04\): \[ 1 + 0.04 = 1.04 \] Next, raise \(1.04\) to the power of \(-15\): \[ (1.04)^{-15} = \frac{1}{(1.04)^{15}} \] **Step 4. Calculate \((1.04)^{15}\):** Using a calculator or logarithms, we find \[ (1.04)^{15} \approx 1.80094 \] Thus, \[ (1.04)^{-15} \approx \frac{1}{1.80094} \approx 0.5551 \] **Step 5. Compute the numerator of the annuity factor:** \[ 1 - (1.04)^{-15} \approx 1 - 0.5551 = 0.4449 \] **Step 6. Divide by the interest rate \(r\):** \[ \frac{0.4449}{0.04} \approx 11.1225 \] **Step 7. Multiply by the annual payment \(P\):** \[ \text{PV} \approx 20,\!000 \times 11.1225 \approx 222,\!450 \] Thus, you need approximately \(\$222,\!450\) in the account at the beginning.