To calculate how much will be in the account after 16 years when Derek makes annual deposits of $2,055.00 starting today, we can use the future value of an annuity formula where the first payment is made immediately (today), and each subsequent payment is made at the end of the year.
Since the first deposit is made today, we treat this as an annuity due. The formula for the future value of an annuity due is:
\[ FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \times (1 + r) \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = annual deposit ($2,055.00)
- \( r \) = annual interest rate (15% or 0.15)
- \( n \) = total number of deposits (16)
Putting the values into the formula, we have:
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Calculate \( (1 + r)^n \): \[ (1 + 0.15)^{16} \approx 6.972 \]
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Calculate \( \left(\frac{(1 + r)^n - 1}{r}\right) \): \[ \frac{(6.972 - 1)}{0.15} = \frac{5.972}{0.15} \approx 39.8133 \]
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Finally, calculate the future value: \[ FV = 2055 \times 39.8133 \times (1 + 0.15) \] \[ FV = 2055 \times 39.8133 \times 1.15 \approx 2055 \times 45.8793 \approx 94,297.55 \]
So, the future value of the account after 16 years will be approximately $94,297.55.
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