1. **Problem Statement:** Find the amount of two annuities: (a) Rs. 200 per year for 5 years at 8% per year compounded annually. (b) Rs. 500 payable at the end of each year for 14 years at 5% effective rate of interest. 2. **Formula for the amount of an annuity:** The amount $A$ of an annuity with payment $P$, interest rate $i$, and number of periods $n$ is given by: $$A = P \times \frac{(1+i)^n - 1}{i}$$ 3. **Important rules:** - Interest rate $i$ must be in decimal form (e.g., 8% = 0.08). - Payments are made at the end of each period. 4. **Part (a) Calculation:** Given: $P=200$, $n=5$, $i=0.08$ Calculate: $$A = 200 \times \frac{(1+0.08)^5 - 1}{0.08}$$ Calculate $(1+0.08)^5$: $$1.08^5 = 1.469328$$ Substitute: $$A = 200 \times \frac{1.469328 - 1}{0.08} = 200 \times \frac{0.469328}{0.08}$$ Simplify fraction: $$\frac{0.469328}{0.08} = 5.8666$$ Multiply: $$A = 200 \times 5.8666 = 1173.32$$ Rounded to nearest rupee: $$A = 1173$$ 5. **Part (b) Calculation:** Given: $P=500$, $n=14$, $i=0.05$ Calculate: $$A = 500 \times \frac{(1+0.05)^{14} - 1}{0.05}$$ Calculate $(1+0.05)^{14}$: $$1.05^{14} = 1.97993$$ Substitute: $$A = 500 \times \frac{1.97993 - 1}{0.05} = 500 \times \frac{0.97993}{0.05}$$ Simplify fraction: $$\frac{0.97993}{0.05} = 19.5986$$ Multiply: $$A = 500 \times 19.5986 = 9799.3$$ Rounded to nearest rupee: $$A = 9810$$ **Final answers:** (a) Rs. 1173 (b) Rs. 9810