Payments with periodic rates. Denise has her heart set on being a millionaire. What payment does Denise need to make at the end of each month over the coming 40 years at \( 9 \% \) APR to reach her retirement goal of \( \$ 1.2 \) million? What monthly payment does Denise need to make to reach her retirement goal? \( \$ \square \) (Round to the nearest cenir)

To determine the monthly payment Denise needs to make to reach her retirement goal of \$1.2 million in 40 years with a 9% annual percentage rate (APR), we'll use the future value of an annuity formula: \[ FV = PMT \times \left(\frac{(1 + r)^n - 1}{r}\right) \] Where: - \( FV \) is the future value (\$1,200,000) - \( PMT \) is the monthly payment - \( r \) is the monthly interest rate (\( \frac{9\%}{12} = 0.0075 \)) - \( n \) is the total number of payments (\( 40 \times 12 = 480 \) months) Rearranging the formula to solve for \( PMT \): \[ PMT = \frac{FV}{\left(\frac{(1 + r)^n - 1}{r}\right)} \] **Step-by-Step Calculation:** 1. **Calculate \((1 + r)^n\):** \[ (1 + 0.0075)^{480} \approx 35.9871 \] 2. **Calculate the annuity factor:** \[ \frac{35.9871 - 1}{0.0075} = \frac{34.9871}{0.0075} \approx 4,664.947 \] 3. **Determine the monthly payment:** \[ PMT = \frac{1,200,000}{4,664.947} \approx 257.48 \] **Conclusion:** Denise needs to make **\$257.48** at the end of each month to reach her retirement goal of \$1.2 million in 40 years at a 9% APR. **Answer:** Denise needs to make a monthly payment of **\$257.48** to reach her retirement goal.