To solve this problem, we will use the formula for compound interest:

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

Where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (in decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested for in years.

Given:

• Principal ($P$) = $3,500
• Annual interest rate ($r$) = 7% = 0.07 (in decimal form)
• Interest is compounded annually ($n$ = 1)
• Time ($t$) = 3 years and 6 years

Let's calculate the amount owed at the end of 3 years and 6 years.

At the end of 3 years:

$A = 3500 \left(1 + \frac{0.07}{1}\right)^{1 \times 3}$ $A = 3500 \left(1 + 0.07\right)^3$ $A = 3500 (1.07)^3$

Calculating $(1.07)^3$: $(1.07)^3 \approx 1.225043$

So, $A \approx 3500 \times 1.225043$ $A \approx 4287.65$

At the end of 6 years:

$A = 3500 \left(1 + \frac{0.07}{1}\right)^{1 \times 6}$ $A = 3500 \left(1 + 0.07\right)^6$ $A = 3500 (1.07)^6$

Calculating $(1.07)^6$: $(1.07)^6 \approx 1.500730$

So, $A \approx 3500 \times 1.500730$ $A \approx 5252.56$

Therefore, the amount owed at the end of 3 years is approximately $4,287.65, and at the end of 6 years, it is approximately $5,252.56.

Would you like any further details or have any questions?

Here are 8 questions related to this problem:

1. How does the frequency of compounding (annually, semi-annually, quarterly) affect the total amount owed?
2. What would the amount be if the interest rate was 5% instead of 7%?
3. How can you use the compound interest formula to calculate the time required to double the investment?
4. What is the difference between simple interest and compound interest?
5. How would the total amount change if the principal amount was $5,000 instead of $3,500?
6. How can you calculate the total interest paid over the 6 years?
7. What would the amount be if interest were compounded monthly?
8. How does the compound interest formula change if additional payments are made periodically?

Tip: Understanding the impact of different interest rates and compounding frequencies can help you make better financial decisions.