1. **State the problem:** Princess deposited 195800 in a savings account paying 2% interest compounded daily (365 days). We need to find the compound interest earned and the future value after 5 years. 2. **Formulas and definitions:** - Interest rate per period: $r = \frac{i}{n}$ where $i$ is the annual interest rate and $n$ is the number of compounding periods per year. - Number of periods: $N = n \times t$ where $t$ is the number of years. - Future value formula for compound interest: $$A = P \left(1 + r\right)^N$$ - Compound interest earned: $$CI = A - P$$ 3. **Identify given values:** - Principal $P = 195800$ - Annual interest rate $i = 0.02$ - Compounding frequency $n = 365$ - Time $t = 5$ years 4. **Calculate interest rate per period:** $$r = \frac{i}{n} = \frac{0.02}{365}$$ 5. **Calculate number of periods:** $$N = n \times t = 365 \times 5 = 1825$$ 6. **Calculate future value:** $$A = 195800 \left(1 + \frac{0.02}{365}\right)^{1825}$$ 7. **Calculate compound interest:** $$CI = A - 195800$$ 8. **Evaluate the expressions:** Calculate $r$: $$r = \frac{0.02}{365} \approx 0.00005479$$ Calculate $A$: $$A = 195800 \left(1 + 0.00005479\right)^{1825}$$ Calculate the base: $$1 + 0.00005479 = 1.00005479$$ Calculate the power: $$1.00005479^{1825} \approx e^{1825 \times 0.00005479} = e^{0.1} \approx 1.10517$$ Calculate $A$: $$A = 195800 \times 1.10517 \approx 216320.23$$ Calculate compound interest: $$CI = 216320.23 - 195800 = 20520.23$$ **Final answers:** - Compound interest earned: 20520.23 - Future value: 216320.23 9. **Compounding frequency:** The compounding frequency is daily, which means interest is compounded 365 times per year.