Suppose that you invest $20,000 in an account that earns interest at an APR of 1.3%, compounded monthly. Determine the accumulated balance after 6 years.

To determine the accumulated balance after 6 years in an account that earns interest compounded monthly, you can use the formula for compound interest:

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

Where:
- $ A $ is the accumulated balance (the amount in the account after interest).
- $ P $ is the principal amount (initial investment).
- $ r $ is the annual interest rate (expressed as a decimal).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the time the money is invested for (in years).

Given:
- $ P = 20,000 $ dollars
- $ r = 1.3\% = 0.013 $
- $ n = 12 $ (since interest is compounded monthly)
- $ t = 6 $ years

Now, plug the values into the formula:

\[
A = 20000 \left(1 + \frac{0.013}{12}\right)^{12 \times 6}
\]

Let's calculate this step by step:

1. Calculate the monthly interest rate:  
   \[
   \frac{0.013}{12} = 0.0010833\overline{3}
   \]

2. Calculate the exponent (number of compounding periods):  
   \[
   12 \times 6 = 72
   \]

3. Calculate the base of the exponentiation:  
   \[
   1 + 0.0010833\overline{3} = 1.0010833\overline{3}
   \]

4. Raise the base to the power of 72:  
   \[
   \left(1.0010833\overline{3}\right)^{72} \approx 1.0797693
   \]

5. Multiply this result by the principal:  
   \[
   A \approx 20000 \times 1.0797693 \approx 21595.39
   \]

So, the accumulated balance after 6 years is approximately **$21,595.39**.

Would you like further details, or do you have any questions?

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Here are 5 related questions you might find interesting:
1. How would the accumulated balance change if the interest was compounded quarterly instead of monthly?
2. What would be the accumulated balance if the investment period was 10 years instead of 6 years?
3. How would the accumulated balance change with a higher APR, say 3%, compounded monthly?
4. What is the formula for continuous compounding, and how would it apply in this scenario?
5. If $5,000 were added to the investment after 3 years, what would the final balance be?

**Tip:** Always double-check the compounding frequency, as it significantly impacts the accumulated interest.

Learn how to calculate the accumulated balance for a $20,000 investment with an APR of 1.3%, compounded monthly, over 6 years. This problem covers the compound interest formula and is suitable for students in Grades 10-12.

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