Find the payment necessary to amortize a \( 4 \% \) loan of \( \$ 1600 \) compounded quarterly, with 16 quarterly payments. The payment size is \( \$ \square \). (Round to the nearest cent.)

To find the payment necessary to amortize a \( 4\% \) loan of \( \$ 1600 \) compounded quarterly with 16 quarterly payments, we can use the formula for the payment on an amortizing loan: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where: - \( M \) is the payment amount, - \( P \) is the principal amount (the loan amount), - \( r \) is the interest rate per period, - \( n \) is the total number of payments. Given: - \( P = 1600 \) - The annual interest rate is \( 4\% \), hence quarterly rate \( r = \frac{0.04}{4} = 0.01 \) - \( n = 16 \) We substitute these values into the formula: \[ M = 1600 \frac{0.01(1 + 0.01)^{16}}{(1 + 0.01)^{16} - 1} \] Calculating \( (1 + 0.01)^{16} \): \[ (1 + 0.01)^{16} \approx 1.1699 \] Now substituting back into the formula: \[ M = 1600 \frac{0.01 \times 1.1699}{1.1699 - 1} \] \[ M = 1600 \frac{0.011699}{0.1699} \] \[ M \approx 1600 \times 0.0688 \approx 110.12 \] Thus, the payment size is \( \$110.12 \).