Questions: [OCR content] Exam #3 Question 10 of 18 (1 point) Question Attempt 1 of 1 Time Remaining: 7 8 9 10 11 12 13 14 15 16 Find the future value of the annuity. Round your answer to the nearest cent. Do not round intermediate steps. Payment Rate Compounded Time 4,000 1.66% Semiannually 3 years The future value of the annuity is Continue

Solution Steps

To find the future value of an annuity, we use the future value of an annuity formula:

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

where:

• \( P \) is the payment amount per period,
• \( r \) is the interest rate per period,
• \( n \) is the total number of periods.

Given that the interest is compounded semiannually, we need to adjust the interest rate and the number of periods accordingly.

We are given the following values for the annuity:

• Payment per period, \( P = 4000 \)
• Annual interest rate, \( r = 0.0166 \)
• Compounding periods per year, \( m = 2 \) (semiannually)
• Total time in years, \( t = 3 \)

The interest rate per period is calculated as: \[ r_p = \frac{r}{m} = \frac{0.0166}{2} = 0.0083 \] The total number of periods is: \[ n = m \times t = 2 \times 3 = 6 \]

Using the future value of an annuity formula: \[ FV = P \times \left( \frac{(1 + r_p)^n - 1}{r_p} \right) \] Substituting the values: \[ FV = 4000 \times \left( \frac{(1 + 0.0083)^6 - 1}{0.0083} \right) \] Calculating the expression gives: \[ FV = 4000 \times \left( \frac{(1.0083)^6 - 1}{0.0083} \right) \approx 24503.55 \]

Final Answer