Find the final amount of money in an account if \( \$ 2,000 \) is deposited at \( 4.5 \% \) interest compounded wee and the money is left for 8 years. The final amount is \( \$ \square \) answer to 2 decimal places

To find the final amount of money in an account with compound interest, we can use the formula for compound interest: \[ A = P \times \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the final amount of money in the account - \( P \) is the principal amount (initial deposit) - \( r \) is the annual interest rate (in decimal form) - \( n \) is the number of times interest is compounded per year - \( t \) is the number of years the money is left in the account Given: - \( P = \$2,000 \) - \( r = 4.5\% = 0.045 \) (in decimal form) - \( n = 52 \) (since interest is compounded weekly) - \( t = 8 \) years Substitute these values into the formula and calculate the final amount \( A \). Calculate the value by following steps: - step0: Calculate: \(2000\left(1+\frac{0.045}{52}\right)^{52\times 8}\) - step1: Divide the terms: \(2000\left(1+\frac{9}{10400}\right)^{52\times 8}\) - step2: Add the terms: \(2000\left(\frac{10409}{10400}\right)^{52\times 8}\) - step3: Multiply the numbers: \(2000\left(\frac{10409}{10400}\right)^{416}\) - step4: Simplify: \(2000\times \frac{10409^{416}}{10400^{416}}\) - step5: Rewrite the expression: \(400\times 5\times \frac{10409^{416}}{10400^{416}}\) - step6: Rewrite the expression: \(400\times 5\times \frac{10409^{416}}{400^{416}\times 26^{416}}\) - step7: Reduce the numbers: \(5\times \frac{10409^{416}}{400^{415}\times 26^{416}}\) - step8: Rewrite the expression: \(5\times \frac{10409^{416}}{25^{415}\times 16^{415}\times 26^{416}}\) - step9: Rewrite the expression: \(5\times \frac{10409^{416}}{5^{830}\times 16^{415}\times 26^{416}}\) - step10: Reduce the numbers: \(\frac{10409^{416}}{5^{829}\times 16^{415}\times 26^{416}}\) The final amount of money in the account after 8 years with a deposit of \$2,000 at 4.5% interest compounded weekly is approximately \$2,866.21.