To solve this problem, we need to calculate how many regular annual payments (denoted as $X$) of $22,820 will be required, along with a special payment of $49,700 in 6 years, to fully repay the loan of $240,203. The interest rate is 9.76% per year, and the first regular payment is made today.

We can approach this as a time value of money problem. Here’s how we will break it down:

1. Step 1: Present value of the regular payments
   The first payment is made today, so it's at time $t = 0$. The rest of the payments are made at the end of each year. We can treat this as a series of regular payments with a known present value (PV).

   We will calculate the present value of the first $X$ regular payments of $22,820 using the present value of an annuity due formula, which accounts for the first payment being made immediately:

   $PV_{\text{regular payments}} = P \times \left( 1 + \frac{1}{r} \right) \times \left[ 1 - \frac{1}{(1 + r)^X} \right]$ where $P$ is the payment amount ($22,820), $r$ is the interest rate per period (9.76% = 0.0976), and $X$ is the number of regular payments.

2. Step 2: Present value of the special payment
   The special payment of $49,700 will be made in 6 years. We need to calculate the present value of this amount using the formula for the present value of a single future payment:

   $PV_{\text{special payment}} = \frac{F}{(1 + r)^6}$ where $F$ is the future value ($49,700) and $r$ is the interest rate (0.0976).

3. Step 3: Total present value
   The total present value of the loan is the sum of the present values of the regular payments and the special payment. This total should be equal to the loan amount ($240,203):

   $PV_{\text{regular payments}} + PV_{\text{special payment}} = 240,203$

We will solve for $X$ by plugging these values into the equations.

Let me calculate this for you.The number of regular annual payments, $X$, needed to repay the loan is approximately 18.73.

Would you like a detailed breakdown of the steps or have any further questions?

Here are five related questions for further exploration:

1. How does changing the interest rate affect the number of payments?
2. What happens if the first regular payment is made one year later instead of today?
3. How is the present value of the special payment calculated if the amount or timing changes?
4. Can the loan be repaid faster by increasing the regular payments slightly?
5. What if the loan term were extended or shortened—how would that impact the total interest paid?

Tip: In time value of money problems, the earlier payments are made, the more impact they have on reducing the loan balance due to the power of compounding interest.