A total of \( \$ 250,000 \) is invested at an annual interest rate of \( 3 \% \). Find the balance in the account after 5 years if it is compounded quarterly (round to the nearest dollar). \( \$ 289,819 \) \( \$ 290,404 \) \( \$ 290,135 \) \( \$ 290,296 \)

To find the balance in the account after 5 years with quarterly compounding, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] where: - \( A \) is the balance in the account after \( t \) years, - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate, - \( n \) is the number of times the interest is compounded per year, - \( t \) is the time in years. Given: - \( P = \$250,000 \), - \( r = 3\% = 0.03 \), - \( n = 4 \) (quarterly compounding), - \( t = 5 \) years. Substitute these values into the formula: \[ A = 250,000 \left( 1 + \frac{0.03}{4} \right)^{4 \times 5} \] Now, we can calculate the balance in the account after 5 years. Calculate the value by following steps: - step0: Calculate: \(250000\left(1+\frac{0.03}{4}\right)^{4\times 5}\) - step1: Divide the terms: \(250000\left(1+\frac{3}{400}\right)^{4\times 5}\) - step2: Add the numbers: \(250000\left(\frac{403}{400}\right)^{4\times 5}\) - step3: Multiply the numbers: \(250000\left(\frac{403}{400}\right)^{20}\) - step4: Simplify: \(250000\times \frac{403^{20}}{400^{20}}\) - step5: Rewrite the expression: \(400\times 625\times \frac{403^{20}}{400^{20}}\) - step6: Reduce the numbers: \(625\times \frac{403^{20}}{400^{19}}\) - step7: Rewrite the expression: \(625\times \frac{403^{20}}{25^{19}\times 16^{19}}\) - step8: Rewrite the expression: \(25^{2}\times \frac{403^{20}}{25^{19}\times 16^{19}}\) - step9: Reduce the numbers: \(1\times \frac{403^{20}}{25^{17}\times 16^{19}}\) - step10: Multiply the fractions: \(\frac{403^{20}}{25^{17}\times 16^{19}}\) The balance in the account after 5 years with quarterly compounding is approximately $290,296. Therefore, the correct answer is $290,296.