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.