Suppose you invest \( \$ 170 \) a month for 6 years into an account earning \( 8 \% \) compounded monthly. After 6 years, you leave the money, without making additional deposits, in the account for another 24 years. How much will you have in the end? \( \$ 93791.36 \) Check Answer

Calculate the value by following steps: - step0: Calculate: \(170\left(\frac{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1\right)}{\left(\frac{0.08}{12}\right)}\right)\) - step1: Remove the parentheses: \(170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\) - step2: Divide the terms: \(170\left(\frac{\left(1+\frac{1}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\) - step3: Add the numbers: \(170\left(\frac{\left(\frac{151}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\) - step4: Multiply the numbers: \(170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{0.08}{12}}\right)\) - step5: Divide the terms: \(170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{1}{150}}\right)\) - step6: Subtract the numbers: \(170\left(\frac{\frac{151^{72}-150^{72}}{150^{72}}}{\frac{1}{150}}\right)\) - step7: Divide the terms: \(170\times \frac{151^{72}-150^{72}}{150^{71}}\) - step8: Rewrite the expression: \(10\times 17\times \frac{151^{72}-150^{72}}{150^{71}}\) - step9: Rewrite the expression: \(10\times 17\times \frac{151^{72}-150^{72}}{10^{71}\times 15^{71}}\) - step10: Reduce the numbers: \(17\times \frac{151^{72}-150^{72}}{10^{70}\times 15^{71}}\) - step11: Multiply the fractions: \(\frac{17\left(151^{72}-150^{72}\right)}{10^{70}\times 15^{71}}\) - step12: Multiply: \(\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}\) Calculate or simplify the expression \( (170 * (((1 + (0.08 / 12))^(12 * 6) - 1) / (0.08 / 12))) * (1 + (0.08 / 12))^(12 * 24) \). Calculate the value by following steps: - step0: Calculate: \(\left(170\left(\frac{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1\right)}{\left(\frac{0.08}{12}\right)}\right)\right)\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 24}\) - step1: Remove the parentheses: \(170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 24}\) - step2: Divide the terms: \(170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(1+\frac{1}{150}\right)^{12\times 24}\) - step3: Add the numbers: \(170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step4: Divide the terms: \(170\left(\frac{\left(1+\frac{1}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step5: Add the numbers: \(170\left(\frac{\left(\frac{151}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step6: Multiply the numbers: \(170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step7: Divide the terms: \(170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{1}{150}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step8: Subtract the numbers: \(170\left(\frac{\frac{151^{72}-150^{72}}{150^{72}}}{\frac{1}{150}}\right)\left(\frac{151}{150}\right)^{12\times 24}\) - step9: Divide the terms: \(170\times \frac{151^{72}-150^{72}}{150^{71}}\times \left(\frac{151}{150}\right)^{12\times 24}\) - step10: Multiply the numbers: \(170\times \frac{151^{72}-150^{72}}{150^{71}}\times \left(\frac{151}{150}\right)^{288}\) - step11: Multiply the terms: \(\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}\times \left(\frac{151}{150}\right)^{288}\) - step12: Evaluate the power: \(\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}\times \frac{151^{288}}{150^{288}}\) - step13: Multiply the fractions: \(\frac{\left(17\times 151^{72}-17\times 150^{72}\right)\times 151^{288}}{10^{70}\times 15^{71}\times 150^{288}}\) - step14: Multiply: \(\frac{17\times 151^{360}-17\times 150^{72}\times 151^{288}}{10^{70}\times 15^{71}\times 150^{288}}\) To determine how much you will have in the end after investing \( \$170 \) a month for 6 years into an account earning \( 8\% \) compounded monthly, we can break the problem into two parts: 1. **Calculate the future value of the monthly investments after 6 years.** 2. **Calculate the future value of that amount after leaving it for another 24 years without additional deposits.** ### Step 1: Future Value of Monthly Investments The formula for the future value of a series of monthly investments (ordinary annuity) is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( P \) = monthly investment (\$170) - \( r \) = monthly interest rate (annual rate / 12) - \( n \) = total number of investments (months) Given: - Annual interest rate = \( 8\% \) or \( 0.08 \) - Monthly interest rate \( r = \frac{0.08}{12} \) - Total months \( n = 6 \times 12 = 72 \) Now, substituting the values into the formula: \[ FV = 170 \times \frac{(1 + \frac{0.08}{12})^{72} - 1}{\frac{0.08}{12}} \] Calculating this gives: \[ FV \approx 15644.31 \] ### Step 2: Future Value After 24 More Years Now, we take the amount from Step 1 and let it grow for another 24 years (or 288 months) without additional deposits. The formula for the future value of a single sum is: \[ FV = PV \times (1 + r)^n \] Where: - \( PV \) = present value from Step 1 - \( r \) = monthly interest rate - \( n \) = total number of months (24 years = 288 months) Substituting the values: \[ FV = 15644.31 \times (1 + \frac{0.08}{12})^{288} \] Calculating this gives: \[ FV \approx 106031.40 \] ### Conclusion After performing the calculations, the final amount in the account after 30 years (6 years of deposits and 24 years of growth) is approximately \( \$106031.40 \). Thus, the answer \( \$93791.36 \) is incorrect. The correct final amount is approximately \( \$106031.40 \).