We are given a loan amortization problem, and we need to find the number of regular annual payments $X$. The known values are:

• Loan amount $P = 240,203$
• Regular annual payment $A = 22,820$
• Special payment in year 6 $S = 49,700$
• Interest rate $i = 9.76\%$ per year
• First payment is made immediately (i.e., it's an annuity due).

Steps to solve:

1. Present Value of the Loan (PV): The loan amount is equivalent to the present value (PV) of both the regular payments and the special payment. Since the first regular payment is made immediately, we are dealing with an annuity due for the regular payments and a future lump sum for the special payment in year 6.

2. Formula for Annuity Due:

   The present value of an annuity due can be calculated using: $PV_{\text{annuity due}} = A \times \left[ \frac{1 - (1 + i)^{-X}}{i} \right] \times (1 + i)$ This formula accounts for the fact that the payments are made at the beginning of each period.

3. Present Value of Special Payment:

   The special payment made in year 6 is a future value discounted back to the present using: $PV_{\text{special}} = S \times (1 + i)^{-6}$

4. Equating Total Present Value:

   The total present value of both the annuity and the special payment must equal the loan amount. Therefore, we have the equation: $240,203 = A \times \left[ \frac{1 - (1 + i)^{-X}}{i} \right] \times (1 + i) + S \times (1 + i)^{-6}$ We will now solve for $X$.

Let me calculate it.The number of regular annual payments $X$ is approximately 18.73 years.

Would you like more details on this calculation or any further clarifications?

Here are 5 related questions to deepen your understanding:

1. How would the number of payments change if the interest rate increased?
2. What if the first regular payment wasn't made today, but in a year?
3. How would adding another special payment affect the number of payments?
4. What is the present value of just the special payment?
5. How would the result change if the regular payments were higher?

Tip: For loans, annuities due (payments made at the beginning of each period) typically result in lower interest costs over time compared to ordinary annuities.