To calculate Justin's annuity savings over the course of 25 years with monthly contributions, we can use the formula for the future value of an annuity:
\[ S = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) \]
Where:
- \(S\) = future value of the annuity
- \(P\) = payment amount per period (monthly contribution)
- \(r\) = annual interest rate (as a decimal)
- \(n\) = number of times interest is compounded per year
- \(t\) = number of years
In this case:
- \(P = 208\) (monthly contribution)
- \(r = 0.05\) (5% annual interest rate)
- \(n = 12\) (interest is compounded monthly)
- \(t = 25\) (25 years)
Now, let's plug in the values into the formula:
\[ S = 208 \times \left( \frac{(1 + \frac{0.05}{12})^{12 \times 25} - 1}{\frac{0.05}{12}} \right) \]
Calculating step-by-step:
- Calculate \(r/n\):
\[ r/n = \frac{0.05}{12} \approx 0.0041667 \]
- Calculate \(nt\):
\[ nt = 12 \times 25 = 300 \]
- Calculate \((1 + r/n)^{nt}\):
\[ (1 + 0.0041667)^{300} \approx 4.4867 \]
- Now substitute back into the formula:
\[ S = 208 \times \left( \frac{4.4867 - 1}{0.0041667} \right) \]
- Calculate \((4.4867 - 1) / 0.0041667\):
\[ (4.4867 - 1) = 3.4867 \]
\[ \frac{3.4867}{0.0041667} \approx 836.008 \]
- Finally, multiply by \(P\):
\[ S \approx 208 \times 836.008 \approx 173,054.66 \]
Therefore, the future value of Justin's annuity savings after 25 years would be approximately $173,054.66.
None of the provided options exactly match the calculation, suggesting there may have been a misunderstanding or an error in the responses provided.