We can solve this problem by calculating the number of regular payments (X) using the present value of annuities and lump-sum formulas.

Key information:

• Loan amount (Present value, $PV$) = $240,203
• Regular payment = $22,820
• Special payment in 6 years = $49,700
• Interest rate $i = 9.76\%$ or 0.0976 per year
• First regular payment is made today, which implies an annuity due.

Step-by-Step Process:

1. Adjust the present value (PV) for the special payment:
   The special payment will be made in 6 years, so we need to find its present value, discounted at the interest rate.

   The formula for present value of a lump sum is: $PV_{\text{lump sum}} = \frac{\text{Special Payment}}{(1 + i)^n}$ where $n = 6$ years, $i = 0.0976$, and the special payment is $49,700.

2. Calculate the present value of the regular annuity payments (annuity due):
   The formula for present value of an annuity due is: $PV_{\text{annuity due}} = P \times \left[ \frac{1 - (1 + i)^{-X}}{i} \right] \times (1 + i)$ where $P = 22,820$ (regular payment), $i = 0.0976$, and $X$ is the number of regular payments to be determined.

3. Set up the equation:
   The total present value (loan amount) is the sum of the present values of the special payment and the annuity payments: $PV = PV_{\text{annuity due}} + PV_{\text{lump sum}}$ Plug in the known values and solve for $X$.

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If you'd like, I can guide you through solving this step-by-step manually or attempt to handle simpler calculations without the tool. Let me know what