Mary is planning for retirement. She deposits \( \$ 1200 \) at the end of each year in an ordinary annuity that pays \( 7 \% \) interest, compounded annually. Answer each part. Do not round any intermediate computations nor answers. If necessary, refer to the list of financial formulas. (a) Find the total value of the annuity at the end of the \( 1^{\text {st }} \) year. (b) Find the total value of the annuity at the end of the \( 2^{\text {nd }} \) year. \( \$ \square \) (c) Find the total value of the annuity at the end of the \( 3^{\text {rd }} \) year. \( \$ \square \)

To find the total value of the annuity at the end of each year, we can use the formula for the future value of an ordinary annuity: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] where: - \( FV \) is the future value of the annuity, - \( P \) is the annual deposit amount, - \( r \) is the annual interest rate, - \( n \) is the number of years. Given: - \( P = \$1200 \) - \( r = 7\% = 0.07 \) Let's calculate the total value of the annuity at the end of each year: (a) At the end of the \( 1^{\text{st}} \) year: \[ FV = 1200 \times \left( \frac{(1 + 0.07)^1 - 1}{0.07} \right) \] (b) At the end of the \( 2^{\text{nd}} \) year: \[ FV = 1200 \times \left( \frac{(1 + 0.07)^2 - 1}{0.07} \right) \] (c) At the end of the \( 3^{\text{rd}} \) year: \[ FV = 1200 \times \left( \frac{(1 + 0.07)^3 - 1}{0.07} \right) \] Let's calculate the values. Calculate the value by following steps: - step0: Calculate: \(\frac{1200\left(\left(1+0.07\right)^{1}-1\right)}{0.07}\) - step1: Add the numbers: \(\frac{1200\left(1.07^{1}-1\right)}{0.07}\) - step2: Calculate: \(\frac{1200\left(1.07-1\right)}{0.07}\) - step3: Subtract the numbers: \(\frac{1200\times 0.07}{0.07}\) - step4: Reduce the fraction: \(1200\) Calculate or simplify the expression \( 1200 * ((1 + 0.07)^2 - 1) / 0.07 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1200\left(\left(1+0.07\right)^{2}-1\right)}{0.07}\) - step1: Add the numbers: \(\frac{1200\left(1.07^{2}-1\right)}{0.07}\) - step2: Convert the expressions: \(\frac{1200\left(\left(\frac{107}{100}\right)^{2}-1\right)}{0.07}\) - step3: Subtract the numbers: \(\frac{1200\times \frac{1449}{10000}}{0.07}\) - step4: Multiply the numbers: \(\frac{\frac{4347}{25}}{0.07}\) - step5: Convert the expressions: \(\frac{\frac{4347}{25}}{\frac{7}{100}}\) - step6: Multiply by the reciprocal: \(\frac{4347}{25}\times \frac{100}{7}\) - step7: Reduce the numbers: \(621\times 4\) - step8: Multiply the numbers: \(2484\) Calculate or simplify the expression \( 1200 * ((1 + 0.07)^3 - 1) / 0.07 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1200\left(\left(1+0.07\right)^{3}-1\right)}{0.07}\) - step1: Add the numbers: \(\frac{1200\left(1.07^{3}-1\right)}{0.07}\) - step2: Convert the expressions: \(\frac{1200\left(\left(\frac{107}{100}\right)^{3}-1\right)}{0.07}\) - step3: Subtract the numbers: \(\frac{1200\times \frac{107^{3}-100^{3}}{100^{3}}}{0.07}\) - step4: Multiply the numbers: \(\frac{\frac{3\times 107^{3}-3\times 100^{3}}{2500}}{0.07}\) - step5: Convert the expressions: \(\frac{\frac{3\times 107^{3}-3\times 100^{3}}{2500}}{\frac{7}{100}}\) - step6: Multiply by the reciprocal: \(\frac{3\times 107^{3}-3\times 100^{3}}{2500}\times \frac{100}{7}\) - step7: Reduce the numbers: \(\frac{3\times 107^{3}-3\times 100^{3}}{25}\times \frac{1}{7}\) - step8: Multiply the fractions: \(\frac{3\times 107^{3}-3\times 100^{3}}{25\times 7}\) - step9: Multiply: \(\frac{3\times 107^{3}-3\times 100^{3}}{175}\) The total value of the annuity at the end of each year is as follows: (a) At the end of the \( 1^{\text{st}} \) year: $1200 (b) At the end of the \( 2^{\text{nd}} \) year: $2484 (c) At the end of the \( 3^{\text{rd}} \) year: $3857.88