Use the compound interest formulas \( \mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{r}}{\mathrm{n}}\right)^{\mathrm{nt}} \) and \( \mathrm{A}=\mathrm{Pe} e^{\mathrm{rt}} \) to solve the problem given. Round answers to the nearest cent. Find the accumulated value of an investment of \( \$ 20,000 \) for 5 years at an interest rate of \( 7 \% \) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

To solve for the accumulated value of the investment using the provided formulas, we will apply each compounding method as follows: a. For semiannual compounding (\( n = 2 \)): \[ A = P\left(1+\frac{r}{n}\right)^{nt} = 20000\left(1+\frac{0.07}{2}\right)^{2 \cdot 5} \approx 20000\left(1+0.035\right)^{10} \approx 20000(1.4107) \approx \$28,214.00 \] b. For quarterly compounding (\( n = 4 \)): \[ A = P\left(1+\frac{r}{n}\right)^{nt} = 20000\left(1+\frac{0.07}{4}\right)^{4 \cdot 5} \approx 20000\left(1+0.0175\right)^{20} \approx 20000(1.4859) \approx \$29,718.35 \] c. For monthly compounding (\( n = 12 \)): \[ A = P\left(1+\frac{r}{n}\right)^{nt} = 20000\left(1+\frac{0.07}{12}\right)^{12 \cdot 5} \approx 20000\left(1+0.0058333\right)^{60} \approx 20000(1.4907) \approx \$29,814.00 \] d. For continuous compounding: \[ A = Pe^{rt} = 20000e^{0.07 \cdot 5} \approx 20000e^{0.35} \approx 20000(1.4191) \approx \$28,382.00 \] So, your accumulated values after 5 years at a 7% interest rate are approximately: - Semiannually: \$28,214.00 - Quarterly: \$29,718.35 - Monthly: \$29,814.00 - Continuously: \$28,382.00