1. **State the problem:** We have a $1000 deposit with an APR of 5.5% compounded continuously. We want to find the balance after 1, 5, and 20 years, and also find the APY. 2. **Formula for continuous compounding:** $$A = P e^{rt}$$ where: - $A$ is the amount after time $t$ - $P$ is the principal (initial deposit) - $r$ is the annual interest rate (as a decimal) - $t$ is the time in years - $e$ is Euler's number (approximately 2.71828) 3. **Calculate balances:** - Given $P=1000$, $r=0.055$ **After 1 year:** $$A = 1000 \times e^{0.055 \times 1} = 1000 \times e^{0.055}$$ Calculate $e^{0.055} \approx 1.05656$ $$A \approx 1000 \times 1.05656 = 1056.56$$ **After 5 years:** $$A = 1000 \times e^{0.055 \times 5} = 1000 \times e^{0.275}$$ Calculate $e^{0.275} \approx 1.31656$ $$A \approx 1000 \times 1.31656 = 1316.56$$ **After 20 years:** $$A = 1000 \times e^{0.055 \times 20} = 1000 \times e^{1.1}$$ Calculate $e^{1.1} \approx 3.00417$ $$A \approx 1000 \times 3.00417 = 3004.17$$ 4. **Calculate APY (Annual Percentage Yield):** APY is given by: $$\text{APY} = \left(e^r - 1\right) \times 100\%$$ Calculate: $$e^{0.055} - 1 \approx 1.05656 - 1 = 0.05656$$ $$\text{APY} \approx 0.05656 \times 100 = 5.66\%$$ **Final answers:** - Balance after 1 year: $1056.56$ - Balance after 5 years: $1316.56$ - Balance after 20 years: $3004.17$ - APY: $5.66\%$