Question

Derek will deposit $2,953.00 per year for 26.00 years into an account that earns 4.00%. The first deposit is made next year. How much will be in the account 26.0 years from today?

Answer 1

To determine how much will be in Derek's account after 26 years, we can use the future value of an ordinary annuity formula. The formula is given by:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

where:
- \( FV \) = future value of the annuity
- \( P \) = annual deposit (payment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of deposits (years)

In this case:
- \( P = 2953 \) (annual deposit)
- \( r = 0.04 \) (4% interest rate)
- \( n = 26 \) (number of years/deposits)

Now, we can plug in the values:

1. Calculate \( (1 + r) \):
\[
1 + r = 1 + 0.04 = 1.04
\]

2. Raise \( (1 + r) \) to the power of \( n \):
\[
(1.04)^{26}
\]

Calculating \( (1.04)^{26} \):
\[
(1.04)^{26} \approx 2.6667311
\]

3. Now substitute back into the future value formula:
\[
FV = 2953 \times \frac{(2.6667311 - 1)}{0.04}
\]

First, calculate \( (2.6667311 - 1) \):
\[
2.6667311 - 1 = 1.6667311
\]

4. Now divide by \( 0.04 \):
\[
\frac{1.6667311}{0.04} = 41.6682775
\]

5. Finally, multiply by \( P \):
\[
FV = 2953 \times 41.6682775 \approx 123,250.23
\]

Thus, the amount in the account 26 years from today will be approximately **$123,250.23**.