Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period. \( \$ 10,000 \); monthly payments for 10 years; interest rate \( 5 \% \) The payment should be \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)
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To find the payment for an annuity due based on a given future value, we can use the formula for the future value of an annuity due: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r) \] Where: - \( FV \) is the future value of the annuity due, which is \( \$10,000 \) - \( P \) is the payment per period (what we are solving for) - \( r \) is the interest rate per period - \( n \) is the total number of payments. Given: - Annual interest rate = 5% - Monthly interest rate \( r = \frac{5\%}{12} = 0.00416667 \) (approximately) - Total number of payments over 10 years = \( 10 \times 12 = 120 \) Now we can rearrange the formula to solve for \( P \): \[ P = \frac{FV}{\frac{(1 + r)^n - 1}{r} \times (1 + r)} \] Substituting the values in: \[ P = \frac{10000}{\frac{(1 + 0.00416667)^{120} - 1}{0.00416667} \times (1 + 0.00416667)} \] Calculating \( (1 + 0.00416667)^{120} = 1.647009 \) (approximately). Now substituting back: \[ P = \frac{10000}{\frac{1.647009 - 1}{0.00416667} \times 1.00416667} \] \[ = \frac{10000}{\frac{0.647009}{0.00416667} \times 1.00416667} \] Calculating \( \frac{0.647009}{0.00416667} \approx 155.283 \) and multiplying by \( 1.00416667 \): \[ 155.283 \times 1.00416667 \approx 155.8915 \] Finally, calculating \( P \): \[ P = \frac{10000}{155.8915} \approx 64.195 \] Thus, the payment should be approximately \( \$64.20 \) when rounded to the nearest cent.
