1. **Problem statement:** Calculate the future value of an investment with monthly contributions and annual compounding interest. 2. **Formula:** The future value $FV$ with regular contributions is given by combining the compound interest on the initial principal $P$ and the future value of an annuity for the monthly contributions $C$: $$FV = P(1 + r)^t + C \times \frac{(1 + r)^t - 1}{r}$$ where: - $P$ = initial principal - $C$ = monthly contribution - $r$ = annual interest rate (as a decimal) - $t$ = number of years 3. **Important rules:** - Since contributions are monthly but compounding is annual, contributions within a year do not earn interest until the next year. - To approximate, treat contributions as if they are made at the end of each year by summing monthly contributions for the year: $C_{year} = 12 \times C$. 4. **Step-by-step calculation:** - Calculate total yearly contribution: $C_{year} = 12 \times C$ - Use the formula: $$FV = P(1 + r)^t + C_{year} \times \frac{(1 + r)^t - 1}{r}$$ 5. **Explanation:** - The first term $P(1 + r)^t$ is the compound interest on the initial principal. - The second term calculates the future value of the yearly contributions treated as an annuity. - This method approximates monthly contributions with yearly compounding. 6. **Example:** If $P=1000$, $C=100$, $r=0.05$, and $t=3$ years: - $C_{year} = 12 \times 100 = 1200$ - Calculate: $$FV = 1000(1 + 0.05)^3 + 1200 \times \frac{(1 + 0.05)^3 - 1}{0.05}$$ $$= 1000(1.157625) + 1200 \times \frac{0.157625}{0.05}$$ $$= 1157.63 + 1200 \times 3.1525 = 1157.63 + 3783 = 4940.63$$ So, the future value after 3 years is approximately 4940.63.