You just borrowed $161,913. You and your heirs plan to repay this loan by making regular monthly payments of $X forever and a special payment of $29,900 in 4 months. The interest rate on the loan is 3.91 percent per month and the first regular payment will be made in 1 month. What is X?

We can solve this problem using the concept of perpetuity, with some adjustments for the special payment made in 4 months.

### Key details:
- Principal (loan amount) = $161,913
- Interest rate per month = 3.91% = 0.0391
- Special payment in 4 months = $29,900
- Monthly payment = $ X $ (to be determined)
- The loan lasts forever (perpetuity).

### Step-by-Step Approach:

1. **Present Value of Perpetuity**:  
   The loan can be considered as a perpetuity with monthly payments of $ X $, starting from month 1. The present value $ PV_{\infty} $ of a perpetuity is given by:

   \[
   PV_{\infty} = \frac{X}{r}
   \]
   where $ r = 0.0391 $ is the monthly interest rate.

2. **Special Payment Adjustment**:  
   In 4 months, a special payment of $29,900 is made. To account for this in the present value, we discount this payment to today (month 0) using the formula:

   \[
   PV_{\text{special}} = \frac{29,900}{(1 + r)^4}
   \]

3. **Total Present Value**:  
   The total present value of the payments must equal the loan amount. Thus:

   \[
   161,913 = \frac{X}{0.0391} + \frac{29,900}{(1 + 0.0391)^4}
   \]

### Solving for $ X $:
First, calculate the present value of the special payment, then substitute it into the total equation to solve for $ X $.

Let me calculate this now.The regular monthly payment $ X $ is approximately **$5,327.99**.

Would you like more details or have any further questions? Here are five related questions you might consider:

1. How does the interest rate affect the monthly payment in a perpetuity?
2. What would happen to the payment amount if the special payment were higher or lower?
3. How can perpetuities be applied in real-life financial situations?
4. What if the loan were to be paid off over a finite period instead of perpetually?
5. How does the timing of the special payment impact the overall calculation?

**Tip**: Perpetuity payments are sensitive to the interest rate. A higher interest rate results in lower payments, while a lower interest rate increases the required payments.

Solve for the regular monthly payment of a $161,913 loan repaid perpetually with a 3.91% monthly interest rate and a $29,900 special payment in 4 months. Learn how to apply the concept of perpetuity and discounting future payments.

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