Questions: You deposit 4000 each year into an account earning 4% annual interest compounded annually. How much will you have in the account in 25 years? (Your answer needs to be to the nearest cent.)

Solution Steps

To solve this problem, we need to calculate the future value of an annuity. The formula for the future value of an annuity compounded annually is:

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

where:

• \( P \) is the annual deposit (\$4000),
• \( r \) is the annual interest rate (4\% or 0.04),
• \( n \) is the number of years (25).

We will use this formula to compute the future value.

We are given:

• Annual deposit, \( P = \$4000 \)
• Annual interest rate, \( r = 0.04 \)
• Number of years, \( n = 25 \)

The future value of an annuity compounded annually is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]

Substituting \( P = 4000 \), \( r = 0.04 \), and \( n = 25 \) into the formula: \[ FV = 4000 \times \frac{(1 + 0.04)^{25} - 1}{0.04} \]

First, calculate \( (1 + 0.04)^{25} \): \[ (1 + 0.04)^{25} \approx 2.6658 \]

Next, compute the numerator: \[ 2.6658 - 1 = 1.6658 \]

Then, divide by the interest rate: \[ \frac{1.6658}{0.04} \approx 41.645 \]

Finally, multiply by the annual deposit: \[ 4000 \times 41.645 \approx 166583.63 \]

Final Answer