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?


This problem involves determining the regular annual payment $ X $ for a loan of \$240,203. The loan will be repaid with regular annual payments of $ X $, plus a special payment of \$49,700 in 6 years. The interest rate is 9.76% per year, and the first payment is made today (which makes it an annuity due).

### Step 1: Break down the loan repayment

The repayment consists of:
1. Regular annual payments of $ X $ for 6 years, with the first payment made today. This is an **annuity due**.
2. A special payment of \$49,700 at the end of 6 years.

### Step 2: Present Value of Regular Payments

Since this is an **annuity due**, the formula for the present value of the regular payments is:

\[
PV_{\text{regular}} = X \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]

Where:
- $ r = 0.0976 $ (annual interest rate),
- $ n = 6 $ (number of years/payments),
- $ X $ is the regular payment.

### Step 3: Present Value of Special Payment

The special payment of \$49,700 will be made at the end of 6 years, so we need to discount it to its present value. The present value of a single future payment is given by:

\[
PV_{\text{special}} = \frac{49,700}{(1 + r)^6}
\]

### Step 4: Equate the Total Present Value to the Loan Amount

The total present value of the regular payments and the special payment must equal the loan amount:

\[
PV_{\text{regular}} + PV_{\text{special}} = 240,203
\]

Now, we can plug in the numbers and solve for $ X $.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the steps to solve it manually or clarify any details. Let me know how you'd like to proceed!

Here are some related questions you might want to explore:
1. How does a special lump-sum payment affect the overall loan calculation?
2. What is the difference between annuity due and ordinary annuity payments in loan repayments?
3.

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?

Learn how to calculate regular annual payments for a loan of $240,203 with a special payment of $49,700 in 6 years and a 9.76% interest rate. This solution covers loan repayment, annuities, and present value concepts. Suitable for college students or advanced high school learners.

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