To find the future value of an investment with different compounding periods, we will use different formulas for each scenario. Let's start by defining the given values:

• Principal (initial investment) $P = \$4500$
• Annual interest rate $r = 4.15\% = 0.0415$
• Time $t = 3$ years

(a) Compounded Annually

The formula for future value with annual compounding is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

• $n$ is the number of times interest is compounded per year (for annual compounding, $n = 1$)

Substituting the values, we get: $A = 4500 \left(1 + \frac{0.0415}{1}\right)^{1 \cdot 3}$ $A = 4500 \left(1 + 0.0415\right)^3$ $A = 4500 (1.0415)^3$

Now, we calculate $(1.0415)^3$: $(1.0415)^3 \approx 1.129158$

So, $A = 4500 \times 1.129158$ $A \approx 5081.21$

Therefore, the future value of the investment when interest is compounded annually is $5081.21.

Would you like details or have any questions? Here are 5 questions for you to ask:

1. How do you calculate the future value with monthly compounding?
2. What is the formula for future value with daily compounding?
3. How do you determine the future value when interest is compounded continuously?
4. Can you explain how to find the doubling time for the given interest rate?
5. What are the key differences between different compounding intervals?

Tip: Always double-check your calculations to ensure accuracy, especially when working with exponents and percentages.