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: \[ FV = P \frac{(1 + r)^n - 1}{r} \] where \( FV \) is the future value of the annuity, \( P \) is the annual payment, \( r \) is the interest rate, and \( n \) is the number of years. Given: - \( P = 1200 \) - \( r = 0.07 \) Now, let's solve for each part: **(a)** Total value at the end of the 1st year: \[ FV_1 = 1200 \cdot \frac{(1 + 0.07)^1 - 1}{0.07} = 1200 \cdot \frac{1.07 - 1}{0.07} = 1200 \cdot \frac{0.07}{0.07} = 1200 \] So, the total value of the annuity at the end of the \( 1^{\text{st}} \) year is \( \$1200 \). **(b)** Total value at the end of the 2nd year: \[ FV_2 = 1200 \cdot \frac{(1 + 0.07)^2 - 1}{0.07} = 1200 \cdot \frac{1.1449 - 1}{0.07} = 1200 \cdot \frac{0.1449}{0.07} \approx 1200 \cdot 2.070 \approx 2484 \] So, the total value of the annuity at the end of the \( 2^{\text{nd}} \) year is \( \$2484 \). **(c)** Total value at the end of the 3rd year: \[ FV_3 = 1200 \cdot \frac{(1 + 0.07)^3 - 1}{0.07} = 1200 \cdot \frac{1.225043 - 1}{0.07} = 1200 \cdot \frac{0.225043}{0.07} \approx 1200 \cdot 3.215 \approx 3858 \] So, the total value of the annuity at the end of the \( 3^{\text{rd}} \) year is \( \$3858 \). In summary: (a) \( \$1200 \) (b) \( \$2484 \) (c) \( \$3858 \)