At the time of her grandson's birth, a grandmother deposits $1000 in an account that pays 2% 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'll use the formula for compound interest:

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

Where:

• \( A \) = the future value of the investment/loan, including interest
• \( P \) = the principal investment amount (the initial deposit or loan amount) = $1000
• \( r \) = the annual interest rate (decimal) = 2% = 0.02
• \( n \) = the number of times that interest is compounded per year = 12 (monthly)
• \( t \) = the number of years the money is invested or borrowed for = 21 years

Now let’s plug in the values into the formula:

\[ A = 1000 \left(1 + \frac{0.02}{12}\right)^{12 \times 21} \]

Calculating \( \frac{0.02}{12} \):

\[ \frac{0.02}{12} = 0.00166667 \]

Now substitute this back into the formula:

\[ A = 1000 \left(1 + 0.00166667\right)^{12 \times 21} \]

Calculating \( 12 \times 21 \):

\[ 12 \times 21 = 252 \]

So now we have:

\[ A = 1000 \left(1.00166667\right)^{252} \]

Next, we can calculate \( \left(1.00166667\right)^{252} \):

Using a calculator:

\[ 1.00166667^{252} \approx 1.490665 \]

Finally, we multiply this by the principal amount:

\[ A \approx 1000 \times 1.490665 \approx 1490.67 \]

Thus, the value of the account at the child's twenty-first birthday will be approximately $1490.67.