Future value (with changing years). Dixie Bank offers a certificate of deposit with an option to select your own investment period. Jonathan has \( \$ 7,500 \) for his CD investment. If the bank is offering a \( 5 \% \) interest rate, compounded annually, how much will the CD be worth at maturity if Jonathan picks a a. three-year investment period? b. five-year investment period? c. ten-year investment period? d. twenty-year investment period?

To calculate the future value of the certificate of deposit (CD) with a changing investment period, we can use the formula for compound interest: \[ FV = PV \times (1 + r)^n \] where: - \( FV \) is the future value of the investment - \( PV \) is the present value of the investment (initial amount) - \( r \) is the annual interest rate - \( n \) is the number of years the money is invested Given: - Initial amount (present value) = $7,500 - Annual interest rate = 5% = 0.05 - We need to calculate the future value for different investment periods: a. Three-year investment period b. Five-year investment period c. Ten-year investment period d. Twenty-year investment period Let's calculate the future value for each investment period: Calculate the value by following steps: - step0: Calculate: \(7500\left(1+0.05\right)^{3}\) - step1: Add the numbers: \(7500\times 1.05^{3}\) - step2: Convert the expressions: \(7500\left(\frac{21}{20}\right)^{3}\) - step3: Evaluate the power: \(7500\times \frac{9261}{8000}\) - step4: Multiply: \(\frac{138915}{16}\) a. For a three-year investment period, the future value of the CD will be $8682.1875. Now, let's calculate the future value for the remaining investment periods: Calculate the value by following steps: - step0: Calculate: \(7500\left(1+0.05\right)^{5}\) - step1: Add the numbers: \(7500\times 1.05^{5}\) - step2: Convert the expressions: \(7500\left(\frac{21}{20}\right)^{5}\) - step3: Simplify: \(7500\times \frac{21^{5}}{20^{5}}\) - step4: Rewrite the expression: \(20\times 375\times \frac{21^{5}}{20^{5}}\) - step5: Reduce the numbers: \(375\times \frac{21^{5}}{20^{4}}\) - step6: Rewrite the expression: \(125\times 3\times \frac{21^{5}}{20^{4}}\) - step7: Rewrite the expression: \(125\times 3\times \frac{21^{5}}{5^{4}\times 4^{4}}\) - step8: Rewrite the expression: \(5^{3}\times 3\times \frac{21^{5}}{5^{4}\times 4^{4}}\) - step9: Reduce the numbers: \(3\times \frac{21^{5}}{5\times 4^{4}}\) - step10: Multiply: \(\frac{3\times 21^{5}}{1280}\) b. For a five-year investment period, the future value of the CD will be $9572.11171875. Now, let's calculate the future value for the remaining investment periods: Calculate the value by following steps: - step0: Calculate: \(7500\left(1+0.05\right)^{10}\) - step1: Add the numbers: \(7500\times 1.05^{10}\) - step2: Convert the expressions: \(7500\left(\frac{21}{20}\right)^{10}\) - step3: Simplify: \(7500\times \frac{21^{10}}{20^{10}}\) - step4: Rewrite the expression: \(20\times 375\times \frac{21^{10}}{20^{10}}\) - step5: Reduce the numbers: \(375\times \frac{21^{10}}{20^{9}}\) - step6: Rewrite the expression: \(125\times 3\times \frac{21^{10}}{20^{9}}\) - step7: Rewrite the expression: \(125\times 3\times \frac{21^{10}}{5^{9}\times 4^{9}}\) - step8: Rewrite the expression: \(5^{3}\times 3\times \frac{21^{10}}{5^{9}\times 4^{9}}\) - step9: Reduce the numbers: \(3\times \frac{21^{10}}{5^{6}\times 4^{9}}\) - step10: Multiply the fractions: \(\frac{3\times 21^{10}}{15625\times 4^{9}}\) c. For a ten-year investment period, the future value of the CD will be approximately $12216.709701. Now, let's calculate the future value for the remaining investment period: Calculate the value by following steps: - step0: Calculate: \(7500\left(1+0.05\right)^{20}\) - step1: Add the numbers: \(7500\times 1.05^{20}\) - step2: Convert the expressions: \(7500\left(\frac{21}{20}\right)^{20}\) - step3: Simplify: \(7500\times \frac{21^{20}}{20^{20}}\) - step4: Rewrite the expression: \(20\times 375\times \frac{21^{20}}{20^{20}}\) - step5: Reduce the numbers: \(375\times \frac{21^{20}}{20^{19}}\) - step6: Rewrite the expression: \(125\times 3\times \frac{21^{20}}{20^{19}}\) - step7: Rewrite the expression: \(125\times 3\times \frac{21^{20}}{5^{19}\times 4^{19}}\) - step8: Rewrite the expression: \(5^{3}\times 3\times \frac{21^{20}}{5^{19}\times 4^{19}}\) - step9: Reduce the numbers: \(3\times \frac{21^{20}}{5^{16}\times 4^{19}}\) - step10: Multiply the fractions: \(\frac{3\times 21^{20}}{5^{16}\times 4^{19}}\) d. For a twenty-year investment period, the future value of the CD will be approximately $19899.732789. Therefore, the future values of the CD for different investment periods are: a. Three-year investment period: $8682.1875 b. Five-year investment period: $9572.11171875 c. Ten-year investment period: $12216.709701 d. Twenty-year investment period: $19899.732789