Question

At the time of her grandson's birth, a grandmother deposits $4000 in an account that pays 5% compounded monthly. What will be the value of the account at the child's twenty-first birthday, assuming that no other deposits or withdrawals are made during this period?
The value of the account will be

Answer 1

To calculate the future value of the account at the child's twenty-first birthday, we can use the formula for compound interest:

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

where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial deposit or investment).
- \( r \) = the annual interest rate (decimal).
- \( n \) = the number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested or borrowed for.

Here are the values we have:
- \( P = 4000 \) (the initial deposit)
- \( r = 0.05 \) (5% interest rate)
- \( n = 12 \) (interest is compounded monthly)
- \( t = 21 \) (the number of years until the child’s twenty-first birthday)

Now, substituting these values into the formula:

\[
A = 4000 \left(1 + \frac{0.05}{12}\right)^{12 \times 21}
\]

Calculating this step-by-step:

1. Calculate \( \frac{0.05}{12} \):
\[
\frac{0.05}{12} = 0.00416667
\]

2. Calculate \( 12 \times 21 \):
\[
12 \times 21 = 252
\]

3. Now calculate \( \left(1 + 0.00416667\right)^{252} \):
\[
1 + 0.00416667 = 1.00416667
\]
\[
(1.00416667)^{252} \approx 2.844161
\]

4. Finally, multiply this by 4000:
\[
A \approx 4000 \times 2.844161 \approx 11376.644
\]

Therefore, the value of the account at the child's twenty-first birthday will be approximately:

\[
\boxed{11376.64}
\]