This problem involves solving for the number of regular payments $X$ in a loan repayment schedule. The situation described involves a loan with regular payments, a lump sum (special payment) in the 6th year, and an initial payment made today. The loan is repaid with annual payments and a specific interest rate.

Let's break it down step by step:

Information provided:

• Loan amount (PV) = $240,203
• Annual regular payment = $22,820
• Special lump sum payment at year 6 = $49,700
• Interest rate (r) = 9.76% = 0.0976 per year
• First payment occurs today = This means the payments are in the form of an annuity due (payments are made at the beginning of each period).

Approach:

We need to calculate the number of regular payments, $X$, such that the present value (PV) of all payments (both regular and special) equals the loan amount.

1. Present value of regular payments:

Since the payments are made at the beginning of each period (annuity due), the formula for the present value of annuity due is:

$PV_{\text{annuity}} = P \times \left(1 + r\right) \times \left[1 - (1 + r)^{-X}\right] \div r$

Where:

• $P = 22,820$ is the regular payment.
• $r = 0.0976$ is the annual interest rate.
• $X$ is the number of regular payments, which we are solving for.

2. Present value of special lump sum payment in year 6:

The lump sum payment occurs in year 6, so its present value at time 0 is:

$PV_{\text{lump sum}} = \frac{49,700}{(1 + r)^6}$

3. Total present value:

The total present value is the sum of the present value of regular payments and the present value of the lump sum, which should equal $240,203:

$PV_{\text{annuity}} + PV_{\text{lump sum}} = 240,203$

We can now solve this equation for $X$.

Let's calculate it.The number of regular annual payments $X$ required to repay the loan is approximately $18.73$. Rounding to two decimal places, the solution is:

$X = 18.73$

This means you would make approximately 18.73 regular payments of $22,820, alongside the special payment of $49,700 in year 6.

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How is the present value of an annuity due different from an ordinary annuity?
2. What if the interest rate were compounded monthly instead of annually?
3. How would the loan term change if you made larger regular payments?
4. What is the significance of making the first payment today (annuity due)?
5. How can you determine the total interest paid over the life of the loan?

Tip: When dealing with loans, understanding the difference between annuity due and ordinary annuity is crucial, as the timing of payments impacts the present value significantly.