**Problem Statement:** Sophie saves 200 at the end of each month. The bank pays 0.250% interest compounded monthly. Find the amount of money she will have at the end of 6 years. 1. Identify given values: - Periodic payment, $R = 200$ - Interest rate per period, $r = 0.0025$ (since 0.250% = 0.0025 as a decimal) - Number of compounding periods per year, $m = 12$ - Total time in years, $t = 6$ 2. The formula for future value $F$ when payments are made periodically is: $$ F = R \left[\frac{(1 + \frac{r}{m})^{mt} - 1}{\frac{r}{m}}\right] $$ 3. Calculate the periodic interest rate and total number of payments: - $\frac{r}{m} = \frac{0.0025}{12} = 0.0002083333$ - $mt = 12 \times 6 = 72$ 4. Calculate the term $(1 + \frac{r}{m})^{mt}$: $$ (1 + 0.0002083333)^{72} = (1.0002083333)^{72} $$ Using approximation or a calculator: $$ (1.0002083333)^{72} \approx e^{72 \times 0.0002083333} = e^{0.015} \approx 1.015113 $$ 5. Substitute values into formula: $$ F = 200 \times \frac{1.015113 - 1}{0.0002083333} = 200 \times \frac{0.015113}{0.0002083333} $$ $$ F = 200 \times 72.496 = 14,499.20 $$ 6. **Answer:** At the end of 6 years, Sophie will have approximately **14,499.20** in her account.