Find the future value for the ordinary annuity with the given payment and interest rate. PMT \( =\$ 2,250 ; 1.40 \% \) compounded quarterly for 8 years. The future value of the ordinary annuity is \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

To find the future value of an ordinary annuity, we can use the formula: \[ FV = PMT \times \frac{(1 + r/n)^{nt} - 1}{r/n} \] Where: - \( FV \) = future value of the annuity - \( PMT \) = payment amount per period - \( r \) = annual interest rate (as a decimal) - \( n \) = number of compounding periods per year - \( t \) = number of years Given: - \( PMT = 2250 \) - \( r = 1.40\% = 0.014 \) - \( n = 4 \) (since it is compounded quarterly) - \( t = 8 \) Now, we can substitute these values into the formula step by step. 1. Calculate \( r/n \): \[ r/n = \frac{0.014}{4} \] 2. Calculate \( nt \): \[ nt = 4 \times 8 \] 3. Calculate \( (1 + r/n)^{nt} \): \[ (1 + r/n)^{nt} = \left(1 + \frac{0.014}{4}\right)^{32} \] 4. Finally, substitute everything back into the future value formula. Let's perform these calculations step by step. Calculate the value by following steps: - step0: Calculate: \(4\times 8\) - step1: Multiply the numbers: \(32\) Calculate or simplify the expression \( (1 + 0.014/4)^(4*8) \). Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.014}{4}\right)^{4\times 8}\) - step1: Divide the terms: \(\left(1+\frac{7}{2000}\right)^{4\times 8}\) - step2: Add the numbers: \(\left(\frac{2007}{2000}\right)^{4\times 8}\) - step3: Multiply the numbers: \(\left(\frac{2007}{2000}\right)^{32}\) - step4: Use the properties of exponents: \(\frac{2007^{32}}{2000^{32}}\) Calculate or simplify the expression \( 0.014/4 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.014}{4}\) - step1: Convert the expressions: \(\frac{\frac{7}{500}}{4}\) - step2: Multiply by the reciprocal: \(\frac{7}{500}\times \frac{1}{4}\) - step3: Multiply the fractions: \(\frac{7}{500\times 4}\) - step4: Multiply: \(\frac{7}{2000}\) Calculate or simplify the expression \( 2250 * ((1 + 0.0035)^(32) - 1) / 0.0035 \). Calculate the value by following steps: - step0: Calculate: \(\frac{2250\left(\left(1+0.0035\right)^{32}-1\right)}{0.0035}\) - step1: Add the numbers: \(\frac{2250\left(1.0035^{32}-1\right)}{0.0035}\) - step2: Convert the expressions: \(\frac{2250\left(\left(\frac{2007}{2000}\right)^{32}-1\right)}{0.0035}\) - step3: Subtract the numbers: \(\frac{2250\times \frac{2007^{32}-2000^{32}}{2000^{32}}}{0.0035}\) - step4: Multiply the numbers: \(\frac{\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}}{0.0035}\) - step5: Convert the expressions: \(\frac{\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}}{\frac{7}{2000}}\) - step6: Multiply by the reciprocal: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}\times \frac{2000}{7}\) - step7: Rewrite the expression: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}\times \frac{250\times 8}{7}\) - step8: Reduce the numbers: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{30}\times 8^{32}}\times \frac{8}{7}\) - step9: Rewrite the expression: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{30}\times 125^{30}\times 8^{32}}\times \frac{8}{7}\) - step10: Rewrite the expression: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{30}\times 125^{30}\times 8^{32}}\times \frac{2^{3}}{7}\) - step11: Reduce the numbers: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{27}\times 125^{30}\times 8^{32}}\times \frac{1}{7}\) - step12: Multiply the fractions: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{123}\times 125^{30}\times 7}\) - step13: Multiply: \(\frac{9\times 2007^{32}-9\times 2000^{32}}{7\times 2^{123}\times 125^{30}}\) Now that we have completed the calculations, let's summarize the results: 1. The value of \( r/n \) is approximately \( 0.0035 \). 2. The value of \( nt \) is \( 32 \). 3. The value of \( (1 + r/n)^{nt} \) is approximately \( 1.118294 \). Now, we can substitute these values into the future value formula: \[ FV = 2250 \times \frac{(1.118294 - 1)}{0.0035} \] Calculating this gives us: \[ FV \approx 2250 \times \frac{0.118294}{0.0035} \approx 2250 \times 33.57085714 \approx 76046.248094 \] Finally, rounding to the nearest cent, the future value of the ordinary annuity is: \[ FV \approx \$ 76,046.25 \] Thus, the future value of the ordinary annuity is \( \$ 76,046.25 \).