Suppose that \( \$ 14,000 \) is deposited for six years at \( 5 \% \) APR. Calculate the interest earned if interest is compounded quarterly. Round your answer to the nearest cent.

To calculate the interest earned on a deposit of $14,000 for six years at 5% APR compounded quarterly, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (initial deposit). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times the interest is compounded per year. - \( t \) is the time the money is invested for in years. Given: - \( P = \$14,000 \) - \( r = 5\% = 0.05 \) (in decimal form) - \( n = 4 \) (compounded quarterly) - \( t = 6 \) years Substitute the given values into the formula: \[ A = 14000 \left(1 + \frac{0.05}{4}\right)^{4 \times 6} \] Now, we can calculate the amount of money accumulated after 6 years, including interest, and then find the interest earned by subtracting the principal amount from the accumulated amount. Calculate the value by following steps: - step0: Calculate: \(14000\left(1+\frac{0.05}{4}\right)^{4\times 6}\) - step1: Divide the terms: \(14000\left(1+\frac{1}{80}\right)^{4\times 6}\) - step2: Add the numbers: \(14000\left(\frac{81}{80}\right)^{4\times 6}\) - step3: Multiply the numbers: \(14000\left(\frac{81}{80}\right)^{24}\) - step4: Simplify: \(14000\times \frac{81^{24}}{80^{24}}\) - step5: Rewrite the expression: \(80\times 175\times \frac{81^{24}}{80^{24}}\) - step6: Reduce the numbers: \(175\times \frac{81^{24}}{80^{23}}\) - step7: Rewrite the expression: \(25\times 7\times \frac{81^{24}}{80^{23}}\) - step8: Rewrite the expression: \(25\times 7\times \frac{81^{24}}{5^{23}\times 16^{23}}\) - step9: Rewrite the expression: \(5^{2}\times 7\times \frac{81^{24}}{5^{23}\times 16^{23}}\) - step10: Reduce the numbers: \(7\times \frac{81^{24}}{5^{21}\times 16^{23}}\) - step11: Multiply the fractions: \(\frac{7\times 81^{24}}{5^{21}\times 16^{23}}\) The amount of money accumulated after 6 years, including interest, is approximately $18,862.91. To find the interest earned, we subtract the principal amount from the accumulated amount: Interest earned = Accumulated amount - Principal amount Interest earned = $18,862.91 - $14,000 Interest earned = $4,862.91 Therefore, the interest earned on a deposit of $14,000 for six years at 5% APR compounded quarterly is approximately $4,862.91.