Let's solve the problem step by step. --- ## Problem Restatement **You expect to live 25 years after retirement. You want to withdraw $75,000 each year, at the beginning of each year. If your retirement account earns 6% per year, how much do you need in your account at retirement?** (This is a present value of an annuity-due problem.) --- ## Step 1: List Given Information - **Number of years (n):** 25 - **Withdrawal per year (PMT):** $75,000 - **Interest rate (r):** 6% (.06) - **Withdrawals at the beginning of each year:** This is an **annuity due**. --- ## Step 2: Formula for Present Value of Annuity Due The present value (PV) of an annuity due: \[ PV = PMT \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r) \] --- ## Step 3: Plug in the Values \[ PV = 75{,}000 \times \left[ \frac{1 - (1 + .06)^{-25}}{.06} \right] \times (1 + .06) \] --- ## Step 4: Calculate Step by Step ### 1. Calculate \((1 + .06)^{-25}\): \[ (1 + .06)^{-25} = (1.06)^{-25} \] \[ \approx .233 \] ### 2. Subtract from 1: \[ 1 - .233 = .767 \] ### 3. Divide by the interest rate: \[ .767 / .06 = 12.7833 \] ### 4. Multiply by \((1 + r)\): \[ 12.7833 \times 1.06 = 13.5503 \] ### 5. Multiply by the withdrawal amount: \[ PV = 75,000 \times 13.5503 = 1,016,272.50 \] --- ## Step 5: Match to Choices The closest answer is **$1,016,273.00** --- ## **Final Answer** \[ \boxed{1,016,273} \] --- ### **Summary Table** | Step | Calculation | Value | |------------------------------|-------------------------|--------------| | \((1.06)^{-25}\) | | .233 | | \(1 - (1.06)^{-25}\) | \(1 - .233\) | .767 | | \(\div .06\) | \(.767 / .06\) | 12.7833 | | \(\times 1.06\) | \(12.7833 \times 1.06\) | 13.5503 | | \(\times 75,000\) | \(13.5503 \times 75,000\)| 1,016,273 | --- If you need the explanation for why we use the annuity due formula, let me know!