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

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

Where:

• $A$ = the future value (the balance in the account)
• $P$ = the principal amount (the initial investment) = $13,000
• $r$ = the annual interest rate (as a decimal) = 4% = 0.04
• $n$ = the number of times the interest is compounded per year = 4 (since it's compounded quarterly)
• $t$ = the number of years = 5

Step-by-step:

1. Substitute the values into the formula:

$A = 13000 \left( 1 + \frac{0.04}{4} \right)^{4 \times 5}$

2. Simplify the terms:

$A = 13000 \left( 1 + 0.01 \right)^{20}$

$A = 13000 \left( 1.01 \right)^{20}$

3. Calculate $(1.01)^{20}$:

$(1.01)^{20} \approx 1.22019$

4. Multiply by the principal:

$A = 13000 \times 1.22019 \approx 15,862.47$

Thus, the balance in the account after 5 years is $15,862.47.

Would you like more details or have any questions? Here are 5 related questions to explore:

1. What if the interest was compounded monthly instead of quarterly?
2. How would the balance change if the investment duration was 10 years instead of 5?
3. What would the final amount be with an APR of 6% instead of 4%?
4. Can you calculate the interest earned (total minus initial investment)?
5. How does continuous compounding differ from quarterly compounding?

Tip: Compounding more frequently (e.g., monthly or daily) increases the final amount slightly because the interest is applied more often!