To find the future value of an investment with monthly compounding interest, we use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}$

where:

• $A$ is the future value of the investment.
• $P$ is the principal (initial investment amount), which is $4,500.
• $r$ is the annual interest rate (6.5%, or 0.065 as a decimal).
• $n$ is the number of times the interest is compounded per year (12 for monthly compounding).
• $t$ is the number of years the money is invested (3 years).

Plugging in the values:

$A = 4500 \left(1 + \frac{0.065}{12}\right)^{12 \cdot 3}$

1. Calculate the monthly interest rate: $\frac{0.065}{12} \approx 0.0054167$

2. Calculate the exponent: $12 \cdot 3 = 36$

3. Substitute and calculate $A$: $A \approx 4500 \times (1 + 0.0054167)^{36}$ $A \approx 4500 \times (1.0054167)^{36}$ $A \approx 4500 \times 1.218994$ $A \approx 5485.47$

So, the value of the investment after 3 years will be approximately $5,485.47.

Would you like further details on any step, or do you have any questions?

---

Here are some related questions to explore this concept further:

1. What would be the future value if the interest were compounded quarterly instead of monthly?
2. How would the investment value change if the interest rate increased to 7%?
3. What is the difference between simple and compound interest in terms of investment growth?
4. How long would it take to double the investment at 6.5% interest compounded monthly?
5. How does continuous compounding affect the investment compared to monthly compounding?

Tip: For compound interest calculations, always ensure you adjust the rate and time to match the compounding period.