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: \( \$ \) (b) \( 8 \% \) compounded annually: \( \$ \) (c) \( 8 \% \) compounded quarterly: \( \$ \) (d) \( 8 \% \) compounded monthly: \( \$ \) Round your answers to the nearest cent. Question Help: Message instructor

To calculate the total interest earned at the end of 20 years for the given investment and interest rates, we can use the formulas for simple interest and compound interest. ### Simple Interest Formula: \[ \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] ### Compound Interest Formula: \[ \text{Compound Interest} = \text{Principal} \times \left(1 + \frac{\text{Rate}}{100}\right)^{\text{Time}} - \text{Principal} \] Given: - Principal = $26,000 - Rate = 8% - Time = 20 years ### (a) Simple Interest: \[ \text{Simple Interest} = \$26,000 \times 8\% \times 20 \] ### (b) Compounded Annually: \[ \text{Compound Interest} = \$26,000 \times \left(1 + \frac{8}{100}\right)^{20} - \$26,000 \] ### (c) Compounded Quarterly: \[ \text{Compound Interest} = \$26,000 \times \left(1 + \frac{8}{100}\right)^{20 \times 4} - \$26,000 \] ### (d) Compounded Monthly: \[ \text{Compound Interest} = \$26,000 \times \left(1 + \frac{8}{100}\right)^{20 \times 12} - \$26,000 \] Let's calculate the total interest earned for each scenario. 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 \( 26000 * (1 + 0.08/100)^{20} - 26000 \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{100}\right)^{20}-26000\) - step1: Divide the terms: \(26000\left(1+\frac{1}{1250}\right)^{20}-26000\) - step2: Add the numbers: \(26000\left(\frac{1251}{1250}\right)^{20}-26000\) - step3: Simplify: \(26000\times \frac{1251^{20}}{1250^{20}}-26000\) - step4: Rewrite the expression: \(\frac{26000\times 1251^{20}}{1250^{20}}-26000\) - step5: Reduce fractions to a common denominator: \(\frac{26000\times 1251^{20}}{1250^{20}}-\frac{26000\times 1250^{20}}{1250^{20}}\) - step6: Transform the expression: \(\frac{26000\times 1251^{20}-26000\times 1250^{20}}{1250^{20}}\) Calculate or simplify the expression \( 26000 * (1 + 0.08/100)^{20*4} - 26000 \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{100}\right)^{20\times 4}-26000\) - step1: Divide the terms: \(26000\left(1+\frac{1}{1250}\right)^{20\times 4}-26000\) - step2: Add the numbers: \(26000\left(\frac{1251}{1250}\right)^{20\times 4}-26000\) - step3: Multiply the numbers: \(26000\left(\frac{1251}{1250}\right)^{80}-26000\) - step4: Simplify: \(26000\times \frac{1251^{80}}{1250^{80}}-26000\) - step5: Rewrite the expression: \(\frac{26000\times 1251^{80}}{1250^{80}}-26000\) - step6: Reduce fractions to a common denominator: \(\frac{26000\times 1251^{80}}{1250^{80}}-\frac{26000\times 1250^{80}}{1250^{80}}\) - step7: Transform the expression: \(\frac{26000\times 1251^{80}-26000\times 1250^{80}}{1250^{80}}\) Calculate or simplify the expression \( 26000 * (1 + 0.08/100)^{20*12} - 26000 \). Calculate the value by following steps: - step0: Calculate: \(26000\left(1+\frac{0.08}{100}\right)^{20\times 12}-26000\) - step1: Divide the terms: \(26000\left(1+\frac{1}{1250}\right)^{20\times 12}-26000\) - step2: Add the numbers: \(26000\left(\frac{1251}{1250}\right)^{20\times 12}-26000\) - step3: Multiply the numbers: \(26000\left(\frac{1251}{1250}\right)^{240}-26000\) - step4: Simplify: \(26000\times \frac{1251^{240}}{1250^{240}}-26000\) - step5: Rewrite the expression: \(\frac{26000\times 1251^{240}}{1250^{240}}-26000\) - step6: Reduce fractions to a common denominator: \(\frac{26000\times 1251^{240}}{1250^{240}}-\frac{26000\times 1250^{240}}{1250^{240}}\) - step7: Transform the expression: \(\frac{26000\times 1251^{240}-26000\times 1250^{240}}{1250^{240}}\) The total interest earned at the end of 20 years for the given investment and interest rates are: - (a) Simple Interest: $41,600 - (b) Compounded Annually: $41,919.68 - (c) Compounded Quarterly: $17,718.31 - (d) Compounded Monthly: $55,001.15 These values are rounded to the nearest cent.