
b. An account with annual​ compounding, an APR of 2​%, and an initial deposit of ​$217​, after 537 years b. After 537 ​years, the balance obtained by investing ​$217 at a rate of 2​% with annual​ compounding, will be ​$

To calculate the balance of an account after 537 years with an initial deposit of $217, an annual interest rate of 2%, and annual compounding, we use the compound interest formula:

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

Where:
- $A$ is the amount of money accumulated after $n$ years, including interest.
- $P$ is the principal amount (the initial deposit), which is $217.
- $r$ is the annual interest rate (in decimal form), which is 0.02.
- $n$ is the number of times that interest is compounded per year (since it's annual compounding, $n = 1$).
- $t$ is the time the money is invested for, which is 537 years.

Let's calculate this:

\[
A = 217 \left(1 + \frac{0.02}{1}\right)^{1 \times 537}
\]

\[
A = 217 \times \left(1.02\right)^{537}
\]

Now, I'll compute the value.

After 537 years, the balance obtained by investing $217 at an annual interest rate of 2% with annual compounding will be approximately **$9,010,563.88**.

Would you like more details or have any questions about this calculation?

Here are 5 related questions you might find interesting:
1. How does the frequency of compounding affect the final balance?
2. What would the balance be if the interest rate was 3% instead of 2%?
3. How does this calculation change if compounded monthly rather than annually?
4. What is the formula to calculate the number of years required to reach a specific balance?
5. How does the initial deposit amount impact the final balance over time?

**Tip:** Compound interest grows faster over time due to the effect of earning interest on interest. The longer the time period, the greater the impact of compounding.

Learn how to calculate compound interest after 537 years with an initial deposit of $217 at an annual interest rate of 2%. Get a detailed solution using the compound interest formula. Understand how compounding affects savings over a long period. Suitable for students in Grades 10-12.

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