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To solve this problem, we need to apply the concept of **loan amortization** and consider both the regular payments and the special payments. The loan amount is $125,793, the annual interest rate is 5.93%, and there are regular annual payments of $X and one special payment of $22,600 at the 10th year.

### Steps to solve:

1. **Find the present value of the special payment**: The special payment of $22,600 is made at the 10th year. Since the loan has an interest rate of 5.93%, we need to discount the $22,600 to its present value (PV) at year 0:
   \[
   \text{PV of special payment} = \frac{22600}{(1 + 0.0593)^{10}}
   \]

2. **Find the present value of the regular payments**: We need to find the present value of the regular annual payments of $X over 10 years. This follows the formula for the present value of an annuity:
   \[
   \text{PV of regular payments} = X \times \frac{1 - (1 + 0.0593)^{-10}}{0.0593}
   \]

3. **Set the total present value equal to the loan amount**: The sum of the present value of the regular payments and the special payment should equal $125,793, the loan amount:
   \[
   125793 = \left(X \times \frac{1 - (1 + 0.0593)^{-10}}{0.0593}\right) + \frac{22600}{(1 + 0.0593)^{10}}
   \]

4. **Solve for $X$**: We'll calculate the values and solve for $X$.

Let me compute this now.The regular annual payment $X$ is approximately **$15,314.34**.

Do you need further clarification or details about the steps?

Here are 5 related questions to expand on this concept:
1. How would the payment $X$ change if the loan period were extended to 15 years?
2. What would happen if the interest rate were reduced to 4%?
3. How does the timing of the special payment affect the total present value?
4. Can this formula be used for monthly payments instead of annual payments?
5. How does inflation affect loan repayment calculations?

**Tip**: When calculating the present value of future payments, it's important to discount each payment individually if they don't occur at regular intervals.

You just borrowed $125,793. You plan to repay this loan by making X regular annual payments of $10,910 and a special payment of $22,600 in 10 years. The interest rate on the loan is 5.93 percent per year and your first regular payment will be made today. What is X?

Learn how to calculate the regular annual payment amount for a loan of $125,793 with a 5.93% interest rate, including a special payment in the 10th year. This guide covers the time value of money, annuities, and loan amortization concepts, suitable for undergraduate students.

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