To find the future value of the grandmother's deposit at the child's 21st birthday, we can use the compound interest formula:

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

Where:

• $A$ is the future value of the investment/loan, including interest.
• $P$ is the principal investment amount (the initial deposit), which is $11,000.
• $r$ is the annual interest rate (decimal), which is 9.5%, or 0.095.
• $n$ is the number of times that interest is compounded per year, which is 12 (since it's compounded monthly).
• $t$ is the time the money is invested for, in years. In this case, it's 21 years.

Now, let's plug the numbers into the formula:

$A = 11,000 \left(1 + \frac{0.095}{12}\right)^{12 \times 21}$

I'll calculate this for you.

The value of the account at the child's 21st birthday will be approximately $80,241.42, assuming no additional deposits or withdrawals are made during this period.

Would you like a detailed explanation of the steps or any further clarification?

Here are 5 related questions for deeper understanding:

1. How does the frequency of compounding affect the final amount in compound interest?
2. What happens if the interest rate changes during the investment period?
3. Can you calculate the interest earned in the first year and compare it to the last year?
4. How would the future value change if the interest was compounded annually instead of monthly?
5. What if withdrawals are made periodically, how would that affect the total amount?

Tip: The more frequently interest is compounded, the more the investment grows, since interest gets added to the principal more often!