You want to be able to withdraw \( \$ 50,000 \) from your account each year for 25 years after you retire. You expect to retire in 20 years. If your account earns \( 7 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals? Round your answer to the nearest cent.

To determine how much you need to deposit each year until retirement to achieve your retirement goals, we can use the formula for calculating the present value of an annuity: \[ PV = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \] Where: - \( PV \) is the present value (the amount you need to deposit each year until retirement), - \( PMT \) is the annual withdrawal amount (\( \$50,000 \)), - \( r \) is the annual interest rate (\( 7\% \)), - \( n \) is the number of years until retirement (20 years). Given that you want to withdraw \( \$50,000 \) each year for 25 years after retirement, we can calculate the present value of the annuity using the formula above. Let's calculate the present value of the annuity. Calculate the value by following steps: - step0: Calculate: \(\frac{50000\left(1-\left(1+0.07\right)^{-25}\right)}{0.07}\) - step1: Add the numbers: \(\frac{50000\left(1-1.07^{-25}\right)}{0.07}\) - step2: Convert the expressions: \(\frac{50000\left(1-\left(\frac{107}{100}\right)^{-25}\right)}{0.07}\) - step3: Subtract the numbers: \(\frac{50000\times \frac{107^{25}-100^{25}}{107^{25}}}{0.07}\) - step4: Multiply the numbers: \(\frac{\frac{50000\times 107^{25}-50000\times 100^{25}}{107^{25}}}{0.07}\) - step5: Convert the expressions: \(\frac{\frac{50000\times 107^{25}-50000\times 100^{25}}{107^{25}}}{\frac{7}{100}}\) - step6: Multiply by the reciprocal: \(\frac{50000\times 107^{25}-50000\times 100^{25}}{107^{25}}\times \frac{100}{7}\) - step7: Multiply the fractions: \(\frac{\left(50000\times 107^{25}-50000\times 100^{25}\right)\times 100}{107^{25}\times 7}\) - step8: Multiply: \(\frac{5000000\times 107^{25}-5000000\times 100^{25}}{107^{25}\times 7}\) The present value of the annuity is approximately \( \$582,679.16 \). Therefore, you need to deposit approximately \( \$582,679.16 \) each year until retirement to achieve your retirement goals.