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.)

We start with the formula for an amortizing loan payment: \[ \text{Payment} = P\frac{r(1+r)^n}{(1+r)^n - 1} \] where \( P \) is the principal, \( r \) is the interest rate per period, and \( n \) is the total number of payments. **Step 1. Identify the variables.** - The principal is \( P = 1600 \). - The annual interest rate is \( 4\% \), or \( 0.04 \) in decimal. Since the loan is compounded quarterly, the interest rate per quarter is \[ r = \frac{0.04}{4} = 0.01. \] - The number of payments is \( n = 16 \) (16 quarterly payments). **Step 2. Substitute the values into the formula.** \[ \text{Payment} = 1600 \cdot \frac{0.01 \left(1+0.01\right)^{16}}{\left(1+0.01\right)^{16} - 1} \] This simplifies to: \[ \text{Payment} = 1600 \cdot \frac{0.01 \left(1.01\right)^{16}}{\left(1.01\right)^{16} - 1} \] **Step 3. Compute \(\left(1.01\right)^{16}\).** \[ \left(1.01\right)^{16} \approx 1.172578644 \] **Step 4. Substitute and calculate the fraction.** The numerator becomes: \[ 0.01 \cdot 1.172578644 \approx 0.01172578644 \] The denominator is: \[ 1.172578644 - 1 = 0.172578644 \] Thus, the fraction is: \[ \frac{0.01172578644}{0.172578644} \approx 0.067909 \] **Step 5. Multiply by the principal.** \[ \text{Payment} \approx 1600 \times 0.067909 \approx 108.6544 \] **Step 6. Round to the nearest cent.** \[ \text{Payment} \approx \$108.65 \] So, the payment size is \( \$108.65 \).