They wish to raise $78,500 at the end of 2 years, making deposits at the beginning of each semester in an account that pays 22% annually, compounded semiannually. What amount should the deposits be?

Step 1: Determine the number of periods (n):

n = 2 \text{ years} \times 2 \text{ semesters/year} = 4 \text{ semesters}

Step 2: Determine the semiannual interest rate (i):

i = \frac{22\%}{2} = 11\% = 0.11

Step 3: Use the future value of an ordinary annuity formula adjusted for payments at the beginning of each period:

FV = P \left( \frac{(1+i)^n - 1}{i} \right) (1+i)

Step 4: Plug in the known values:

78500 = P \left( \frac{(1+0.11)^4 - 1}{0.11} \right) (1+0.11)

Step 5: Simplify and solve for P:

78500 = P \left( \frac{(1.11)^4 - 1}{0.11} \right) (1.11)

78500 = P \left( \frac{1.518743 - 1}{0.11} \right) (1.11)

78500 = P \left( \frac{0.518743}{0.11} \right) (1.11)

P\approx15015.87