read the question carefully, 

You just borrowed $240,203. You plan to repay this loan by making (X) regular annual payments of $22,820 and a special payment of special payment of $49,700 in 6 years. The interest rate on the loan is 9.76 percent per year and your first regular payment will be made today. What is X?


I now realize that the problem is asking for the **number of regular payments**, not the duration over which the payments are made. The goal is to find $ X $, the **amount of each regular payment**. Let's carefully revisit the solution.

Here’s what we know:

- **Loan amount** $ P = 240,203 $
- **Regular payment** $ X $
- **Interest rate** $ i = 9.76\% $ per year
- **Special payment** $ S = 49,700 $, made in 6 years
- First regular payment is made **immediately** (which makes this an **annuity due**).

To find the correct value of $ X $, we'll follow these steps:

### Steps to Solve:

1. **Present Value of Regular Payments**:
   Since the first payment is made today, we treat the regular payments as an **annuity due**. The present value of an annuity due is calculated as:
   \[
   PV_{\text{annuity due}} = X \times \left[ \frac{1 - (1 + i)^{-5}}{i} \right] \times (1 + i)
   \]
   (Note that the exponent is 5 because only 5 more payments will follow after today, making a total of 6 regular payments).

2. **Present Value of the Special Payment**:
   The special payment of $49,700 is a **future lump sum**, so we discount it to the present:
   \[
   PV_{\text{special}} = 49,700 \times (1 + i)^{-6}
   \]

3. **Total Present Value**:
   The total present value of both the annuity and the special payment must equal the loan amount:
   \[
   240,203 = PV_{\text{annuity due}} + PV_{\text{special}}
   \]

I'll now calculate the regular payment $ X $.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can still guide you through the manual steps to solve this problem, or you can try again when my tools are available. Let me know how you'd like to proceed!

You just borrowed $240,203. You plan to repay this loan by making (X) regular annual payments of $22,820 and a special payment of $49,700 in 6 years. The interest rate on the loan is 9.76 percent per year and your first regular payment will be made today. What is X?

This problem explains how to find the regular payments for a loan of $240,203 with an interest rate of 9.76% and a special payment of $49,700 due in 6 years. Learn about annuities, present value calculations, and solving for unknown payments using the annuity due formula. Suitable for undergraduate finance or economics students.

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