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? \( \$ \) (Round to the nearest cent.)

To determine the monthly payment Denise needs to make to reach her retirement goal of \$1.2 million in 40 years at an annual interest rate of 9%, we can use the future value of an annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity (the retirement goal, \$1,200,000). - \( P \) is the monthly payment (what we want to find). - \( r \) is the monthly interest rate (annual rate divided by 12). - \( n \) is the total number of payments (number of years multiplied by 12). ### Step 1: Calculate the monthly interest rate and total number of payments - Annual interest rate \( = 9\% = 0.09 \) - Monthly interest rate \( r = \frac{0.09}{12} \) - Total number of payments \( n = 40 \times 12 \) ### Step 2: Substitute the values into the formula and solve for \( P \) Rearranging the formula to solve for \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] Now, let's calculate the values step by step. 1. Calculate \( r \) and \( n \): - \( r = \frac{0.09}{12} \) - \( n = 40 \times 12 \) 2. Substitute \( FV = 1,200,000 \), \( r \), and \( n \) into the formula for \( P \). Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(40\times 12\) - step1: Multiply the numbers: \(480\) Calculate or simplify the expression \( 0.09/12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.09}{12}\) - step1: Convert the expressions: \(\frac{\frac{9}{100}}{12}\) - step2: Multiply by the reciprocal: \(\frac{9}{100}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{3}{100}\times \frac{1}{4}\) - step4: Multiply the fractions: \(\frac{3}{100\times 4}\) - step5: Multiply: \(\frac{3}{400}\) Calculate or simplify the expression \( 1200000*(0.0075)*((1+0.0075)^480-1) \). Calculate the value by following steps: - step0: Calculate: \(1200000\times 0.0075\left(\left(1+0.0075\right)^{480}-1\right)\) - step1: Add the numbers: \(1200000\times 0.0075\left(1.0075^{480}-1\right)\) - step2: Convert the expressions: \(1200000\times 0.0075\left(\left(\frac{403}{400}\right)^{480}-1\right)\) - step3: Subtract the numbers: \(1200000\times 0.0075\times \frac{403^{480}-400^{480}}{400^{480}}\) - step4: Multiply the terms: \(9000\times \frac{403^{480}-400^{480}}{400^{480}}\) - step5: Rewrite the expression: \(200\times 45\times \frac{403^{480}-400^{480}}{400^{480}}\) - step6: Rewrite the expression: \(200\times 45\times \frac{403^{480}-400^{480}}{200^{480}\times 2^{480}}\) - step7: Reduce the numbers: \(45\times \frac{403^{480}-400^{480}}{200^{479}\times 2^{480}}\) - step8: Rewrite the expression: \(5\times 9\times \frac{403^{480}-400^{480}}{200^{479}\times 2^{480}}\) - step9: Rewrite the expression: \(5\times 9\times \frac{403^{480}-400^{480}}{25^{479}\times 8^{479}\times 2^{480}}\) - step10: Rewrite the expression: \(5\times 9\times \frac{403^{480}-400^{480}}{5^{958}\times 8^{479}\times 2^{480}}\) - step11: Reduce the numbers: \(9\times \frac{403^{480}-400^{480}}{5^{957}\times 8^{479}\times 2^{480}}\) - step12: Multiply the fractions: \(\frac{9\left(403^{480}-400^{480}\right)}{5^{957}\times 2^{1917}}\) - step13: Multiply: \(\frac{9\times 403^{480}-9\times 400^{480}}{5^{957}\times 2^{1917}}\) Calculate or simplify the expression \( (1+0.0075)^480-1 \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.0075\right)^{480}-1\) - step1: Add the numbers: \(1.0075^{480}-1\) - step2: Convert the expressions: \(\left(\frac{403}{400}\right)^{480}-1\) - step3: Rewrite the expression: \(\frac{403^{480}}{400^{480}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{403^{480}}{400^{480}}-\frac{400^{480}}{400^{480}}\) - step5: Transform the expression: \(\frac{403^{480}-400^{480}}{400^{480}}\) Calculate or simplify the expression \( 1200000*0.0075/35.109902 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1200000\times 0.0075}{35.109902}\) - step1: Multiply the numbers: \(\frac{9000}{35.109902}\) - step2: Convert the expressions: \(\frac{9000}{\frac{17554951}{500000}}\) - step3: Multiply by the reciprocal: \(9000\times \frac{500000}{17554951}\) - step4: Multiply: \(\frac{9000\times 500000}{17554951}\) - step5: Multiply: \(\frac{4500000000}{17554951}\) ### Step 1: Calculate Monthly Interest Rate and Total Payments - Monthly interest rate \( r = 0.0075 \) - Total number of payments \( n = 480 \) ### Step 2: Substitute Values into the Formula We need to calculate the monthly payment \( P \) using the rearranged formula: \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] Substituting the values: - \( FV = 1,200,000 \) - \( r = 0.0075 \) - \( n = 480 \) Calculating \( (1 + r)^n - 1 \): \[ (1 + 0.0075)^{480} - 1 \approx 35.109902 \] Now substituting into the formula for \( P \): \[ P = \frac{1,200,000 \times 0.0075}{35.109902} \approx 256.34 \] ### Final Answer Denise needs to make a monthly payment of approximately **\$256.34** to reach her retirement goal of \$1.2 million in 40 years at a 9% annual interest rate.