Question 12 of 15, Step 1 of 1 A couple deposits \( \$ 21,000 \) into an account earning \( 6 \% \) annual interest for 15 years. Calculate the future value of the investment if the interest is compounded weekly. Round your answer to the nearest cent. Answer How to enter your answer (opens in new window) Kormulas Keyboard Shortcuts

To calculate the future value of an investment with compound interest, we can use the formula: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( FV \) = future value of the investment - \( P \) = principal amount (initial deposit) - \( r \) = annual interest rate (as a decimal) - \( n \) = number of times interest is compounded per year - \( t \) = number of years the money is invested ### Given: - \( P = 21000 \) - \( r = 0.06 \) (6% as a decimal) - \( n = 52 \) (since interest is compounded weekly) - \( t = 15 \) ### Step 1: Substitute the values into the formula Now, we will substitute the known values into the formula: \[ FV = 21000 \left(1 + \frac{0.06}{52}\right)^{52 \times 15} \] ### Step 2: Calculate the expression Let's calculate the future value using the formula. I'll perform the calculation now. Calculate the value by following steps: - step0: Calculate: \(21000\left(1+\left(\frac{0.06}{52}\right)\right)^{52\times 15}\) - step1: Divide the terms: \(21000\left(1+\frac{3}{2600}\right)^{52\times 15}\) - step2: Add the numbers: \(21000\left(\frac{2603}{2600}\right)^{52\times 15}\) - step3: Multiply the numbers: \(21000\left(\frac{2603}{2600}\right)^{780}\) - step4: Simplify: \(21000\times \frac{2603^{780}}{2600^{780}}\) - step5: Rewrite the expression: \(200\times 105\times \frac{2603^{780}}{2600^{780}}\) - step6: Rewrite the expression: \(200\times 105\times \frac{2603^{780}}{200^{780}\times 13^{780}}\) - step7: Reduce the numbers: \(105\times \frac{2603^{780}}{200^{779}\times 13^{780}}\) - step8: Rewrite the expression: \(5\times 21\times \frac{2603^{780}}{200^{779}\times 13^{780}}\) - step9: Rewrite the expression: \(5\times 21\times \frac{2603^{780}}{25^{779}\times 8^{779}\times 13^{780}}\) - step10: Rewrite the expression: \(5\times 21\times \frac{2603^{780}}{5^{1558}\times 8^{779}\times 13^{780}}\) - step11: Reduce the numbers: \(21\times \frac{2603^{780}}{5^{1557}\times 8^{779}\times 13^{780}}\) - step12: Multiply the fractions: \(\frac{21\times 2603^{780}}{5^{1557}\times 8^{779}\times 13^{780}}\) The future value of the investment is approximately \( 51624.87 \). ### Final Answer: The future value of the investment after 15 years, with weekly compounding at an annual interest rate of 6%, is **\$51,624.87**.