Find the final amount of money in an account if \( \$ 5,200 \) is deposited at \( 5.5 \% \) interest compounded semi-annually and the money is left for 6 years.

To find the final amount of money in an account with compound interest, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. Given: - \( P = 5200 \) - \( r = 5.5\% = 0.055 \) - \( n = 2 \) (since the interest is compounded semi-annually) - \( t = 6 \) Now, we can substitute these values into the formula: \[ A = 5200 \left(1 + \frac{0.055}{2}\right)^{2 \times 6} \] Let's calculate the expression step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{0.055}{2} = 0.0275 \] 2. Calculate \( nt \): \[ 2 \times 6 = 12 \] 3. Now substitute these values into the formula: \[ A = 5200 \left(1 + 0.0275\right)^{12} \] 4. Calculate \( 1 + 0.0275 \): \[ 1 + 0.0275 = 1.0275 \] 5. Now raise \( 1.0275 \) to the power of \( 12 \): \[ (1.0275)^{12} \] 6. Finally, multiply by \( 5200 \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(1.0275^{12}\) - step1: Convert the expressions: \(\left(\frac{411}{400}\right)^{12}\) - step2: Use the properties of exponents: \(\frac{411^{12}}{400^{12}}\) Calculate or simplify the expression \( 5200 * (1.0275^12) \). Calculate the value by following steps: - step0: Calculate: \(5200\times 1.0275^{12}\) - step1: Convert the expressions: \(5200\left(\frac{411}{400}\right)^{12}\) - step2: Simplify: \(5200\times \frac{411^{12}}{400^{12}}\) - step3: Rewrite the expression: \(400\times 13\times \frac{411^{12}}{400^{12}}\) - step4: Reduce the numbers: \(13\times \frac{411^{12}}{400^{11}}\) - step5: Multiply: \(\frac{13\times 411^{12}}{400^{11}}\) After performing the calculations, we find: 1. \( (1.0275)^{12} \approx 1.384784 \) 2. Now, we can calculate the final amount \( A \): \[ A \approx 5200 \times 1.384784 \approx 7200.88 \] Thus, the final amount of money in the account after 6 years is approximately **\$7,200.88**.