Gray invested \( \$ 9000 \) four times a year in an annuity due ecurities for a period of 2 years at an interest rate of \( 8 \% \) unded quarterly. Using the ordinary annuity table, calcu alue of the annuity due at the end of the 2 -year period.

To calculate the future value of an annuity due, we can use the formula for the future value of an annuity due, which is given by: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \] Where: - \( FV \) = future value of the annuity due - \( P \) = payment amount per period - \( r \) = interest rate per period - \( n \) = total number of payments ### Given: - \( P = 9000 \) (the amount invested each quarter) - Annual interest rate = \( 8\% \) or \( 0.08 \) - Since the interest is compounded quarterly, the quarterly interest rate \( r \) is: \[ r = \frac{0.08}{4} = 0.02 \] - The total number of payments over 2 years, with 4 payments per year, is: \[ n = 2 \times 4 = 8 \] ### Step 1: Calculate the future value of the annuity due Substituting the values into the formula: \[ FV = 9000 \times \left( \frac{(1 + 0.02)^8 - 1}{0.02} \right) \times (1 + 0.02) \] ### Step 2: Calculate \( (1 + 0.02)^8 \) \[ (1 + 0.02)^8 = 1.02^8 \approx 1.171659 \] ### Step 3: Calculate \( \frac{(1.171659 - 1)}{0.02} \) \[ \frac{(1.171659 - 1)}{0.02} = \frac{0.171659}{0.02} \approx 8.58295 \] ### Step 4: Calculate the future value Now substituting back into the future value formula: \[ FV = 9000 \times 8.58295 \times 1.02 \] Calculating \( 9000 \times 8.58295 \): \[ 9000 \times 8.58295 \approx 77246.55 \] Now multiplying by \( 1.02 \): \[ FV \approx 77246.55 \times 1.02 \approx 78791.86 \] ### Conclusion The future value of the annuity due at the end of the 2-year period is approximately: \[ \boxed{78791.86} \]