You just borrowed $207,416. You and your heirs plan to repay this loan by making regular monthly payments of $X forever and a special payment of $27,100 in 3 months. The interest rate on the loan is 2.66 percent per month and the first regular payment will be made in 1 month. What is X?

To solve for the regular monthly payment $ X $, we'll use the concept of the present value of perpetuity and account for the special payment of $27,100 in 3 months. Here's a step-by-step breakdown:

### Step 1: Present Value of Regular Payments (Perpetuity)
Since the regular payments are planned to continue forever, the formula for the present value of perpetuity is:

\[
PV_{\text{perpetuity}} = \frac{X}{r}
\]

where:
- $ PV_{\text{perpetuity}} $ is the present value of the perpetuity,
- $ X $ is the regular monthly payment,
- $ r $ is the monthly interest rate, which is 2.66% or 0.0266.

### Step 2: Present Value of the Special Payment
The special payment of $27,100 will be made in 3 months, so we need to calculate its present value. The formula for the present value of a lump sum is:

\[
PV_{\text{special}} = \frac{27,100}{(1 + r)^3}
\]

### Step 3: Total Present Value of All Payments
The total amount borrowed ($207,416) should equal the present value of the perpetuity plus the present value of the special payment:

\[
207,416 = \frac{X}{0.0266} + \frac{27,100}{(1 + 0.0266)^3}
\]

### Step 4: Solving for $ X $

First, calculate the present value of the special payment:

\[
PV_{\text{special}} = \frac{27,100}{(1 + 0.0266)^3} \approx \frac{27,100}{1.0814} \approx 25,054.95
\]

Now, substitute into the total present value equation:

\[
207,416 = \frac{X}{0.0266} + 25,054.95
\]

Simplify and solve for $ X $:

\[
207,416 - 25,054.95 = \frac{X}{0.0266}
\]

\[
182,361.05 = \frac{X}{0.0266}
\]

\[
X = 182,361.05 \times 0.0266 \approx 4,852.82
\]

### Conclusion
The regular monthly payment $ X $ is approximately **$4,852.82**.

Would you like more details or further explanation on this process?

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Here are 5 related questions to expand your understanding:
1. How does the interest rate affect the size of the regular payments?
2. What if the special payment was made at a different time? How would the calculation change?
3. How do we adjust the formula for perpetuity if the payments are delayed or irregular?
4. What if the interest rate is compounded differently, such as annually instead of monthly?
5. Can we generalize this method for other types of loans with different payment structures?

**Tip:** When working with perpetuities, always ensure that the interest rate and payment periods are consistent in units (monthly, annually, etc.).

Learn how to calculate the regular monthly payment of $X for a loan of $207,416 with a perpetuity and a special payment of $27,100 due in 3 months. This detailed step-by-step solution covers key finance concepts like perpetuity, present value, and lump sum payments. Ideal for college-level finance and mathematics students.

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