1. **State the problem:** We want to find the future value of an account with an initial deposit of 8500, an interest rate of 3.2% per year, compounded at different frequencies (semi-annually, monthly, daily) over 20 years. 2. **Formula:** The compound interest formula is: $$A(t) = P\left(1 + \frac{r}{n}\right)^{nt}$$ where: - $P$ is the principal (initial deposit), - $r$ is the annual interest rate (decimal), - $n$ is the number of compounding periods per year, - $t$ is the number of years, - $A(t)$ is the amount after $t$ years. 3. **Given values:** - $P = 8500$ - $r = 0.032$ - $t = 20$ 4. **Calculate for each compounding frequency:** **a. Semi-annually ($n=2$):** $$A = 8500\left(1 + \frac{0.032}{2}\right)^{2 \times 20} = 8500\left(1 + 0.016\right)^{40} = 8500(1.016)^{40}$$ Calculate $(1.016)^{40}$: $$1.016^{40} \approx 1.872$$ So, $$A \approx 8500 \times 1.872 = 15912$$ Rounded to nearest dollar: **15912** **b. Monthly ($n=12$):** $$A = 8500\left(1 + \frac{0.032}{12}\right)^{12 \times 20} = 8500\left(1 + 0.0026667\right)^{240} = 8500(1.0026667)^{240}$$ Calculate $(1.0026667)^{240}$: $$1.0026667^{240} \approx 1.896$$ So, $$A \approx 8500 \times 1.896 = 16116$$ Rounded to nearest dollar: **16116** **c. Daily ($n=365$):** $$A = 8500\left(1 + \frac{0.032}{365}\right)^{365 \times 20} = 8500\left(1 + 0.00008767\right)^{7300} = 8500(1.00008767)^{7300}$$ Calculate $(1.00008767)^{7300}$: $$1.00008767^{7300} \approx 1.897$$ So, $$A \approx 8500 \times 1.897 = 16125$$ Rounded to nearest dollar: **16125** **Final answers:** - a. 15912 - b. 16116 - c. 16125