Certainly! Let's solve the problem step-by-step to find the final value of the account after making annual deposits and earning compounded interest over nine years.
Given data:
- Initial deposit at the beginning of each year: [tex]$14,000
- Annual interest rate: 7% (or 0.07 as a decimal)
- Number of years: 9
We will use the formula for the future value of an annuity compounded annually:
\[ A = P \times \left( \frac{{(1 + r)^n - 1}}{r} \right) \]
Where:
- \( A \) is the future value of the annuity,
- \( P \) is the annual deposit,
- \( r \) is the annual interest rate,
- \( n \) is the number of years.
Using our given data:
- \( P = 14,000 \)
- \( r = 0.07 \)
- \( n = 9 \)
Let's plug these values into the formula:
\[ A = 14,000 \times \left( \frac{{(1 + 0.07)^9 - 1}}{0.07} \right) \]
First, calculate \( (1 + 0.07)^9 \):
\[ 1.07^9 \approx 1.838459 \]
Next, subtract 1 from this result:
\[ 1.838459 - 1 = 0.838459 \]
Now, divide by the annual interest rate r:
\[ \frac{0.838459}{0.07} \approx 11.977986 \]
Finally, multiply by the annual deposit P:
\[ 14,000 \times 11.977986 \approx 167,691.84 \]
Therefore, the final value of the account at the end of nine years is approximately \( 167,691.84 \).
From the provided options, the closest match is:
D. $[/tex]167,692
