If \( \$ 26,000 \) is invested in an account for 20 years. Calculate the total interest earned at the end of 20 years if the interest is: (a) \( 8 \% \) simple interest: \( \$ 41,600 \) (b) \( 8 \% \) compounded annually: \( \$ \) (c) \( 8 \% \) compounded quarterly: \( \$ \) (d) \( 8 \% \) compounded monthly: \( \$ \) Round your answers to the nearest cent.

To solve the problem, we will calculate the total interest earned for each case using the appropriate formulas. ### (a) Simple Interest The formula for simple interest is given by: \[ I = P \times r \times t \] where: - \( I \) is the interest earned, - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate (in decimal), - \( t \) is the time in years. Given: - \( P = 26000 \) - \( r = 0.08 \) - \( t = 20 \) Calculating the simple interest: \[ I = 26000 \times 0.08 \times 20 \] ### (b) Compounded Annually The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest, - \( n \) is the number of times that interest is compounded per year. For annual compounding: - \( n = 1 \) Calculating the total amount: \[ A = 26000 \left(1 + \frac{0.08}{1}\right)^{1 \times 20} \] ### (c) Compounded Quarterly For quarterly compounding: - \( n = 4 \) Calculating the total amount: \[ A = 26000 \left(1 + \frac{0.08}{4}\right)^{4 \times 20} \] ### (d) Compounded Monthly For monthly compounding: - \( n = 12 \) Calculating the total amount: \[ A = 26000 \left(1 + \frac{0.08}{12}\right)^{12 \times 20} \] Now, let's perform the calculations for each case. Calculate the value by following steps: - step0: Calculate: \(26000\times 0.08\times 20\) - step1: Multiply the terms: \(2080\times 20\) - step2: Multiply the numbers: \(41600\) Calculate or simplify the expression \( 26000 * (1 + 0.08/1)^(1*20) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{1}\right)^{1\times 20}\) - step1: Divide the terms: \(26000\left(1+\frac{2}{25}\right)^{1\times 20}\) - step2: Add the numbers: \(26000\left(\frac{27}{25}\right)^{1\times 20}\) - step3: Calculate: \(26000\left(\frac{27}{25}\right)^{20}\) - step4: Simplify: \(26000\times \frac{27^{20}}{25^{20}}\) - step5: Rewrite the expression: \(125\times 208\times \frac{27^{20}}{25^{20}}\) - step6: Rewrite the expression: \(5^{3}\times 208\times \frac{27^{20}}{5^{40}}\) - step7: Reduce the numbers: \(208\times \frac{27^{20}}{5^{37}}\) - step8: Multiply: \(\frac{208\times 27^{20}}{5^{37}}\) Calculate or simplify the expression \( 26000 * (1 + 0.08/12)^(12*20) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{12}\right)^{12\times 20}\) - step1: Divide the terms: \(26000\left(1+\frac{1}{150}\right)^{12\times 20}\) - step2: Add the numbers: \(26000\left(\frac{151}{150}\right)^{12\times 20}\) - step3: Multiply the numbers: \(26000\left(\frac{151}{150}\right)^{240}\) - step4: Simplify: \(26000\times \frac{151^{240}}{150^{240}}\) - step5: Rewrite the expression: \(50\times 520\times \frac{151^{240}}{150^{240}}\) - step6: Rewrite the expression: \(50\times 520\times \frac{151^{240}}{50^{240}\times 3^{240}}\) - step7: Reduce the numbers: \(520\times \frac{151^{240}}{50^{239}\times 3^{240}}\) - step8: Rewrite the expression: \(10\times 52\times \frac{151^{240}}{50^{239}\times 3^{240}}\) - step9: Rewrite the expression: \(10\times 52\times \frac{151^{240}}{10^{239}\times 5^{239}\times 3^{240}}\) - step10: Reduce the numbers: \(52\times \frac{151^{240}}{10^{238}\times 5^{239}\times 3^{240}}\) - step11: Rewrite the expression: \(4\times 13\times \frac{151^{240}}{10^{238}\times 5^{239}\times 3^{240}}\) - step12: Rewrite the expression: \(4\times 13\times \frac{151^{240}}{2^{238}\times 5^{238}\times 5^{239}\times 3^{240}}\) - step13: Rewrite the expression: \(2^{2}\times 13\times \frac{151^{240}}{2^{238}\times 5^{238}\times 5^{239}\times 3^{240}}\) - step14: Reduce the numbers: \(13\times \frac{151^{240}}{2^{236}\times 5^{238}\times 5^{239}\times 3^{240}}\) - step15: Multiply the fractions: \(\frac{13\times 151^{240}}{2^{236}\times 5^{477}\times 3^{240}}\) Calculate or simplify the expression \( 26000 * (1 + 0.08/4)^(4*20) \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{4}\right)^{4\times 20}\) - step1: Divide the terms: \(26000\left(1+\frac{1}{50}\right)^{4\times 20}\) - step2: Add the numbers: \(26000\left(\frac{51}{50}\right)^{4\times 20}\) - step3: Multiply the numbers: \(26000\left(\frac{51}{50}\right)^{80}\) - step4: Simplify: \(26000\times \frac{51^{80}}{50^{80}}\) - step5: Rewrite the expression: \(50\times 520\times \frac{51^{80}}{50^{80}}\) - step6: Reduce the numbers: \(520\times \frac{51^{80}}{50^{79}}\) - step7: Rewrite the expression: \(10\times 52\times \frac{51^{80}}{50^{79}}\) - step8: Rewrite the expression: \(10\times 52\times \frac{51^{80}}{10^{79}\times 5^{79}}\) - step9: Reduce the numbers: \(52\times \frac{51^{80}}{10^{78}\times 5^{79}}\) - step10: Rewrite the expression: \(4\times 13\times \frac{51^{80}}{10^{78}\times 5^{79}}\) - step11: Rewrite the expression: \(4\times 13\times \frac{51^{80}}{2^{78}\times 5^{78}\times 5^{79}}\) - step12: Rewrite the expression: \(2^{2}\times 13\times \frac{51^{80}}{2^{78}\times 5^{78}\times 5^{79}}\) - step13: Reduce the numbers: \(13\times \frac{51^{80}}{2^{76}\times 5^{78}\times 5^{79}}\) - step14: Multiply the fractions: \(\frac{13\times 51^{80}}{2^{76}\times 5^{157}}\) Now, let's summarize the results for each case: ### (a) Simple Interest The total interest earned with simple interest is: \[ I = 41600 \] ### (b) Compounded Annually The total amount after 20 years with annual compounding is approximately: \[ A \approx 121184.89 \] Thus, the interest earned is: \[ I = A - P = 121184.89 - 26000 = 95184.89 \] ### (c) Compounded Quarterly The total amount after 20 years with quarterly compounding is approximately: \[ A \approx 126761.42 \] Thus, the interest earned is: \[ I = A - P = 126761.42 - 26000 = 100761.42 \] ### (d) Compounded Monthly The total amount after 20 years with monthly compounding is approximately: \[ A \approx 128096.87 \] Thus, the interest earned is: \[ I = A - P = 128096.87 - 26000 = 102096.87 \] ### Summary of Total Interest Earned - (a) Simple Interest: \( \$ 41,600 \) - (b) Compounded Annually: \( \$ 95,184.89 \) - (c) Compounded Quarterly: \( \$ 100,761.42 \) - (d) Compounded Monthly: \( \$ 102,096.87 \) All amounts are rounded to the nearest cent.