Rob deposits \( \$ 2000 \) into a one-year \( C D \) at a rate of \( 5.6 \% \) interest compounded monthly. 1) What is his ending balance after the year? \( \$ \) 2) How much interest does he eam? \$ 3) What is his APY to two decimals as a percent?

Given: - Initial deposit: $2000 - Interest rate: 5.6% compounded monthly - Time period: 1 year 1) To find the ending balance after the year, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the ending balance - \( P \) 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 time the money is invested for, in years Substitute the given values into the formula: - \( P = 2000 \) - \( r = 0.056 \) - \( n = 12 \) (monthly compounding) - \( t = 1 \) \[ A = 2000 \left(1 + \frac{0.056}{12}\right)^{12 \times 1} \] 2) To find the interest earned, subtract the initial deposit from the ending balance: \[ \text{Interest earned} = A - P \] 3) To find the Annual Percentage Yield (APY), we can use the formula: \[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \] Let's calculate these step by step. Calculate the value by following steps: - step0: Calculate: \(2000\left(1+\frac{0.056}{12}\right)^{12\times 1}\) - step1: Divide the terms: \(2000\left(1+\frac{7}{1500}\right)^{12\times 1}\) - step2: Add the terms: \(2000\left(\frac{1507}{1500}\right)^{12\times 1}\) - step3: Multiply: \(2000\left(\frac{1507}{1500}\right)^{12}\) - step4: Simplify: \(2000\times \frac{1507^{12}}{1500^{12}}\) - step5: Rewrite the expression: \(500\times 4\times \frac{1507^{12}}{1500^{12}}\) - step6: Rewrite the expression: \(500\times 4\times \frac{1507^{12}}{500^{12}\times 3^{12}}\) - step7: Reduce the numbers: \(4\times \frac{1507^{12}}{500^{11}\times 3^{12}}\) - step8: Rewrite the expression: \(4\times \frac{1507^{12}}{4^{11}\times 125^{11}\times 3^{12}}\) - step9: Reduce the numbers: \(\frac{1507^{12}}{4^{10}\times 125^{11}\times 3^{12}}\) 1) The ending balance after the year is approximately $2114.92. Next, let's calculate the interest earned. Calculate the value by following steps: - step0: Calculate: \(2114.919857-2000\) - step1: Subtract the numbers: \(114.919857\) 2) The interest earned is approximately $114.92. Finally, let's calculate the Annual Percentage Yield (APY) to two decimal places as a percent. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.056}{12}\right)^{12}-1\) - step1: Divide the terms: \(\left(1+\frac{7}{1500}\right)^{12}-1\) - step2: Add the terms: \(\left(\frac{1507}{1500}\right)^{12}-1\) - step3: Rewrite the expression: \(\frac{1507^{12}}{1500^{12}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{1507^{12}}{1500^{12}}-\frac{1500^{12}}{1500^{12}}\) - step5: Transform the expression: \(\frac{1507^{12}-1500^{12}}{1500^{12}}\) 3) The Annual Percentage Yield (APY) is approximately 5.75%.