Find the payment necessary to amortize a $4 \%$ loan of $\$ 2000$ compounded quarterly, with 12 quarterly payments.
The payment size is $\$ \square$.
(Round to the nearest cent.)

Step 1 ：The problem is asking for the payment necessary to pay off a loan with a certain interest rate, principal amount, and number of payments. This is a common problem in finance and can be solved using the formula for the payment of an amortizing loan, which is: \(P = \frac{{r*PV}}{{1 - (1 + r)^{-n}}}\) where: P is the payment, r is the interest rate per period, PV is the present value or principal amount of the loan, n is the number of payments.

Step 2 ：In this case, the interest rate per period is 4% per year compounded quarterly, which is 1% per quarter. The principal amount of the loan is $2000, and there are 12 quarterly payments.

Step 3 ：Substitute the given values into the formula: \(r = 0.01\), \(PV = 2000\), \(n = 12\)

Step 4 ：Solve the equation to find the payment size: \(P = 177.7\)

Step 5 ：Final Answer: The payment size is \(\boxed{\$177.70}\)