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:
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Calculate \( (1 + r) \): \[ 1 + r = 1 + 0.04 = 1.04 \]
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Raise \( (1 + r) \) to the power of \( n \): \[ (1.04)^{26} \]
Calculating \( (1.04)^{26} \): \[ (1.04)^{26} \approx 2.6667311 \]
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Now substitute back into the future value formula: \[ FV = 2953 \times \frac{(2.6667311 - 1)}{0.04} \]
First, calculate \( (2.6667311 - 1) \): \[ 2.6667311 - 1 = 1.6667311 \]
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Now divide by \( 0.04 \): \[ \frac{1.6667311}{0.04} = 41.6682775 \]
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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.