You just borrowed $25,858. You plan to repay this loan by making X regular annual payments of $3,170 and a special payment of $24,900 in 10 years. The interest rate on the loan is 14.56 percent per year and your first regular payment will be made today. What is X?

To solve this problem, we can use the **Present Value of an Annuity** formula for the regular payments and the **Present Value of a Lump Sum** formula for the special payment. Let's break it down step by step.

### Key Information:
- Loan amount (Present Value, PV): \$25,858
- Regular annual payment: \$3,170
- Special payment in 10 years: \$24,900
- Interest rate: 14.56% per year (expressed as a decimal: $ r = 0.1456 $)
- First regular payment made today (indicating that the payments are an annuity due)
- The number of regular payments is $ X $

### Step 1: Present Value of Regular Payments (Annuity Due)
An annuity due means that the payments are made at the beginning of each period. The formula for the Present Value of an Annuity Due is:

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

Where:
- $ P = 3,170 $ (regular annual payment)
- $ r = 0.1456 $ (interest rate per year)
- $ X $ is the number of regular payments

### Step 2: Present Value of the Lump Sum
The special payment of \$24,900 is made in 10 years. We can use the Present Value of a Lump Sum formula to calculate its present value today:

\[
PV_{\text{lump sum}} = \frac{FV}{(1 + r)^N}
\]

Where:
- $ FV = 24,900 $ (future value)
- $ r = 0.1456 $ (interest rate)
- $ N = 10 $ (number of years)

### Step 3: Total Present Value
The total present value of the loan is the sum of the present value of the regular payments and the present value of the lump sum. This should equal the loan amount:

\[
PV_{\text{loan}} = PV_{\text{annuity due}} + PV_{\text{lump sum}}
\]

Now let's calculate $ X $. I'll compute the values for you.The number of regular annual payments $ X $ is approximately 11.15, which suggests you will make 11 regular payments of \$3,170.

Would you like more details on the calculation steps? 

Here are 5 related questions:
1. What is the formula for the Present Value of an Annuity Due?
2. How does an annuity due differ from an ordinary annuity?
3. What happens if the interest rate changes during the loan term?
4. How would the result change if the special payment was made at a different time?
5. Can this method be used for monthly payments instead of annual ones?

**Tip:** Always round off the number of payments to the next whole number, as fractional payments typically don't exist in loan scenarios.

This solution explains how to calculate the regular annual payments for a loan of $25,858, with a special payment of $24,900 due in 10 years. Using the Present Value of an Annuity Due and Lump Sum formulas, find out how many payments are needed to repay the loan at an interest rate of 14.56%.

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