This is a loan repayment problem with an immediate payment, where we need to determine the number of annual regular payments (denoted as $X$) under a given interest rate and loan terms. Here's a breakdown of the process:

Key information:

• Loan amount ($P$) = $240,203
• Annual payment ($A$) = $22,820
• Special payment after 6 years = $49,700
• Interest rate ($r$) = 9.76% per year = 0.0976
• First payment is made today (which implies this is an annuity due).

To find $X$, we need to follow these steps:

1. Adjust the payments for time value:

   • The first regular payment occurs today, meaning it's an "annuity due" (payments at the start of the period).
   • The special payment after 6 years is handled separately, as it comes after the series of regular payments.

2. Use the formula for present value of an annuity due: The present value $PV$ of an annuity due can be calculated using the formula: $PV = A \times \left( \frac{1 - (1 + r)^{-X}}{r} \right) \times (1 + r)$ where $A$ is the payment, $r$ is the interest rate, and $X$ is the number of payments.

3. Handle the special payment separately: The present value of the special payment is calculated as: $PV_{\text{special}} = \frac{49,700}{(1 + r)^6}$ because the special payment occurs at the end of the 6th year.

4. Set up the equation to solve for $X$: The sum of the present value of the annuity due and the present value of the special payment should equal the loan amount: $240,203 = A \times \left( \frac{1 - (1 + r)^{-X}}{r} \right) \times (1 + r) + PV_{\text{special}}$

I will now compute the value of $X$.The number of regular annual payments ($X$) required to repay the loan, given the provided terms, is approximately 18.73.

Would you like further details or clarification on any part of the solution?

Here are five related questions to expand your understanding:

1. How is the present value of annuities calculated for different payment schedules?
2. What adjustments are necessary for annuities with immediate payments versus end-of-period payments?
3. How does changing the interest rate affect the number of payments required?
4. Can the formula used here be applied to monthly payments, and if so, what changes are needed?
5. What would happen if the special payment were made earlier or later in the loan term?

Tip: When solving loan-related problems, always adjust for the timing of payments—whether they're made at the beginning or the end of each period—as this impacts the calculations significantly.