After 538 ​years, the balance obtained by investing ​$222 at a rate of 2​% with annual​ compounding, will be ​$
  
enter your response here.

To calculate the balance after 538 years with annual compounding, we can use the formula for compound interest:

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

Where:
- $ A $ is the final amount (balance).
- $ P = 222 $ is the principal (initial investment).
- $ r = 0.02 $ is the annual interest rate (2%).
- $ n = 1 $ is the number of times the interest is compounded per year (annually).
- $ t = 538 $ is the time in years.

Substituting the values:

\[
A = 222 \left(1 + \frac{0.02}{1}\right)^{1 \times 538}
\]

\[
A = 222 \left(1.02\right)^{538}
\]

Now, I will compute the value of $ A $.The balance after 538 years of investing $222 at a rate of 2% with annual compounding will be approximately **$9,402,544.17**.

Would you like further details on this calculation or have any questions?

Here are five related questions to expand on this topic:
1. What is the difference between annual and continuous compounding?
2. How would the balance change if the interest rate was higher, say 5%?
3. What would be the effect of compounding more frequently, such as quarterly?
4. How do you calculate the time required for an investment to double at a specific interest rate?
5. What is the formula for calculating compound interest with monthly compounding?

**Tip:** Increasing the compounding frequency (e.g., quarterly or monthly) leads to a slightly higher final balance compared to annual compounding at the same interest rate.

After 538 years, the balance obtained by investing ​$222 at a rate of 2​% with annual​ compounding, will be ​$

Learn how to calculate the final balance after 538 years for an initial investment of $222 with a 2% interest rate compounded annually. This problem demonstrates the use of the compound interest formula and explores exponential growth in finance. Suitable for high school students in Grades 9-12.

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