15. INVESTMENTS Determine the amount of an investment if \( \$ 1000 \) is invested at an interest rate of \( 8 \% \) compounded quarterly for 2 years.

To determine the amount of an investment, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the amount of the investment after \( t \) years, - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate (in decimal form), - \( n \) is the number of times the interest is compounded per year, - \( t \) is the time the money is invested for in years. Given: - Principal amount (\( P \)) = $1000 - Annual interest rate (\( r \)) = 8% = 0.08 - Compounded quarterly, so \( n = 4 \) - Time (\( t \)) = 2 years Substitute the given values into the formula: \[ A = 1000 \left( 1 + \frac{0.08}{4} \right)^{4 \times 2} \] Now, we can calculate the amount of the investment. Calculate the value by following steps: - step0: Calculate: \(1000\left(1+\frac{0.08}{4}\right)^{4\times 2}\) - step1: Divide the terms: \(1000\left(1+\frac{1}{50}\right)^{4\times 2}\) - step2: Add the numbers: \(1000\left(\frac{51}{50}\right)^{4\times 2}\) - step3: Multiply the numbers: \(1000\left(\frac{51}{50}\right)^{8}\) - step4: Simplify: \(1000\times \frac{51^{8}}{50^{8}}\) - step5: Rewrite the expression: \(50\times 20\times \frac{51^{8}}{50^{8}}\) - step6: Reduce the numbers: \(20\times \frac{51^{8}}{50^{7}}\) - step7: Rewrite the expression: \(10\times 2\times \frac{51^{8}}{50^{7}}\) - step8: Rewrite the expression: \(10\times 2\times \frac{51^{8}}{10^{7}\times 5^{7}}\) - step9: Reduce the numbers: \(2\times \frac{51^{8}}{10^{6}\times 5^{7}}\) - step10: Rewrite the expression: \(2\times \frac{51^{8}}{2^{6}\times 5^{6}\times 5^{7}}\) - step11: Reduce the numbers: \(1\times \frac{51^{8}}{2^{5}\times 5^{6}\times 5^{7}}\) - step12: Multiply the fractions: \(\frac{51^{8}}{500000\times 5^{7}}\) The amount of the investment after 2 years with an interest rate of 8% compounded quarterly is approximately $1171.66.