Use the appropriate compound interest formula to compute the balance in the account after the stated period of time
​$13 comma 000 is invested for 5 years with an APR of 4​% and quarterly compounding.
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Part 1
The balance in the account after 5 years is ​$
  
enter your response here.
​(Round to the nearest cent as​ needed.)

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!

Use the appropriate compound interest formula to compute the balance in the account after the stated period of time: $13,000 is invested for 5 years with an APR of 4% and quarterly compounding. The balance in the account after 5 years is ​$____. (Round to the nearest cent as needed.)

Learn how to compute the compound interest for a $13,000 investment over 5 years with a 4% APR and quarterly compounding. Using the formula A = P(1 + r/n)^(nt), this solution demonstrates the step-by-step process, rounding to the nearest cent. Ideal for high school students studying compound interest and exponential growth.

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