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?

Question

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:

\[ \boxed{11376.64} \]