1. **Problem statement:** Find the amount of an annuity immediate of Rs. 100 payable half-yearly for 15 years at an interest rate of 4% per annum compounded half-yearly. 2. **Identify given data:** - Payment per period, $R = 100$ - Number of years, $t = 15$ - Compounding periods per year, $m = 2$ (half-yearly) - Nominal annual interest rate, $i_{nom} = 4\%$ 3. **Convert the annual interest rate to the interest rate per period:** $$ i = \frac{i_{nom}}{m} = \frac{4\%}{2} = 2\% = 0.02 $$ 4. **Calculate the total number of payment periods:** $$ n = m \times t = 2 \times 15 = 30 $$ 5. **Apply the formula for the amount of an annuity immediate:** $$ A = R \times \frac{(1+i)^n - 1}{i} $$ Plugging in the values: $$ A = 100 \times \frac{(1+0.02)^{30} - 1}{0.02} $$ 6. **Calculate powers and simplify:** $$ (1.02)^{30} \approx 1.811364 $$ So, $$ A = 100 \times \frac{1.811364 - 1}{0.02} = 100 \times \frac{0.811364}{0.02} = 100 \times 40.5682 = 4056.82 $$ 7. **Answer:** The amount of the annuity immediate after 15 years is approximately Rs. 4056.82.