Determine the amount needed such that when it comes time for retirement, an individual can make monthly withdraws in the amount of \( \$ 2,154 \) for 30 years from an account paying \( 5.1 \% \) compounded monthly. Round your answer to the nearest cent. a. \( \$ 396,721.78 \) b. \( \$ 398,407.85 \) c. \( \$ 775,440 \) d. \( \$ 1,833,962.40 \)

To determine the amount needed for retirement, we can use the formula for the future value of an annuity: \[ FV = P \times \left( \frac{{(1 + r)^n - 1}}{r} \right) \] where: - \( FV \) is the future value of the annuity, - \( P \) is the monthly withdrawal amount, - \( r \) is the monthly interest rate, - \( n \) is the number of periods. Given: - Monthly withdrawal amount, \( P = \$2,154 \), - Monthly interest rate, \( r = 5.1\% = 0.051 \), - Number of periods, \( n = 30 \times 12 = 360 \) (since there are 12 months in a year and 30 years). Substitute the values into the formula and calculate the future value of the annuity. Calculate the value by following steps: - step0: Calculate: \(\frac{2154\left(\left(1+0.051\right)^{360}-1\right)}{0.051}\) - step1: Add the numbers: \(\frac{2154\left(1.051^{360}-1\right)}{0.051}\) - step2: Convert the expressions: \(\frac{2154\left(\left(\frac{1051}{1000}\right)^{360}-1\right)}{0.051}\) - step3: Subtract the terms: \(\frac{2154\times \frac{1051^{360}-1000^{360}}{1000^{360}}}{0.051}\) - step4: Multiply the numbers: \(\frac{\frac{1077\times 1051^{360}-1077\times 1000^{360}}{2^{1079}\times 125^{360}}}{0.051}\) - step5: Convert the expressions: \(\frac{\frac{1077\times 1051^{360}-1077\times 1000^{360}}{2^{1079}\times 125^{360}}}{\frac{51}{1000}}\) - step6: Multiply by the reciprocal: \(\frac{1077\times 1051^{360}-1077\times 1000^{360}}{2^{1079}\times 125^{360}}\times \frac{1000}{51}\) - step7: Rewrite the expression: \(\frac{3\left(359\times 1051^{360}-359\times 1000^{360}\right)}{2^{1079}\times 125^{360}}\times \frac{1000}{51}\) - step8: Reduce the numbers: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{2^{1079}\times 125^{360}}\times \frac{1000}{17}\) - step9: Rewrite the expression: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{2^{1079}\times 125^{360}}\times \frac{8\times 125}{17}\) - step10: Rewrite the expression: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{2^{1079}\times 125^{360}}\times \frac{2^{3}\times 125}{17}\) - step11: Reduce the numbers: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{2^{1076}\times 125^{359}}\times \frac{1}{17}\) - step12: Multiply the fractions: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{2^{1076}\times 125^{359}\times 17}\) - step13: Multiply: \(\frac{359\times 1051^{360}-359\times 1000^{360}}{17\times 2^{1076}\times 125^{359}}\) The future value of the annuity is approximately \( \$2.53 \times 10^{12} \). Now, we need to round this amount to the nearest cent to match the options provided. Let's round the amount to the nearest cent. Round the number by following steps: - step0: Round \(2.52728e+12\) to \(1\) decimal places using standard rounding method: \(2.52728e+12\) - step1: Round \(2.52728e+12\) to \(1\) decimal places using standard rounding method\(:\) \(2.5\) The amount needed for retirement, when rounded to the nearest cent, is approximately \$2.5 billion. Therefore, the closest option provided is: a. \$396,721.78