To find the amount owed on the loan with compound interest, we use the formula for compound interest:

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

where:

• $A$ is the amount owed after time $t$,
• $P$ is the principal amount (initial loan amount),
• $r$ is the annual interest rate (expressed as a decimal),
• $n$ is the number of times interest is compounded per year,
• $t$ is the time the money is invested or borrowed for, in years.

In this problem:

• $P = 7500$ dollars,
• $r = 0.07$ (7% expressed as a decimal),
• $n = 1$ (interest is compounded once per year).

(a) Amount owed at the end of 1 year

Here, $t = 1$: $A_1 = 7500 \left(1 + \frac{0.07}{1}\right)^{1 \cdot 1}$ $A_1 = 7500 (1 + 0.07)$ $A_1 = 7500 \times 1.07$ $A_1 = 8025$

So, the amount owed at the end of 1 year is $\$8025$.

(b) Amount owed at the end of 2 years

Here, $t = 2$: $A_2 = 7500 \left(1 + \frac{0.07}{1}\right)^{1 \cdot 2}$ $A_2 = 7500 (1 + 0.07)^2$ $A_2 = 7500 \times 1.07^2$ $A_2 = 7500 \times 1.1449$ $A_2 = 8586.75$

So, the amount owed at the end of 2 years is $\$8586.75$.

Would you like more details on this process or have any other questions?

Here are some additional questions related to this topic:

1. How would the amount owed change if the interest rate was 5% instead of 7%?
2. What would be the amount owed after 5 years with the same 7% interest rate?
3. How does the frequency of compounding (e.g., semi-annually, quarterly) affect the total amount owed?
4. What is the formula for continuously compounded interest?
5. How can you calculate the effective annual rate (EAR) from the nominal rate?
6. What are some practical implications of compound interest in personal finance?
7. How does compound interest compare to simple interest over long periods?
8. What strategies can borrowers use to minimize the amount of interest paid on loans?

Tip: When dealing with compound interest, always ensure that the interest rate and the compounding periods are correctly matched in your calculations to avoid errors.