You have an investment worth $51,937 that is expected to make regular monthly payments of $1,396 for 12 months and a special payment of $X in 10 months. The expected return for the investment is 0.67 percent per month and the first regular payment will be made today. What is X? Round final answer to the nearest dollar.

This problem can be solved using the concept of the **Present Value of an Annuity** and a **Lump-Sum Payment**.

The given information is:
- Initial investment $ P_0 = 51,937 $
- Regular monthly payments $ PMT = 1,396 $
- Number of monthly payments $ n = 12 $
- The special payment of $ X $ is made in month 10.
- Interest rate $ r = 0.67\% $ per month = 0.0067
- The first regular payment is made today.

### Step-by-step process:

1. **Present Value of Regular Payments (Annuity):**
   Since the first payment is made today, this is an **annuity due**.

   The formula for the present value of an annuity due is:
   \[
   PV_{\text{annuity}} = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
   \]
   where:
   - $ PMT = 1,396 $
   - $ r = 0.0067 $
   - $ n = 12 $

2. **Present Value of Special Payment (X):**
   The special payment $ X $ is made in 10 months, so we need to discount it back to the present. The present value of the lump sum is:
   \[
   PV_{\text{X}} = \frac{X}{(1 + r)^{10}}
   \]

3. **Total Present Value:**
   The total present value must be equal to the initial investment $ 51,937 $:
   \[
   PV_{\text{annuity}} + PV_{\text{X}} = 51,937
   \]

Let's calculate $ PV_{\text{annuity}} $ first and then solve for $ X $.The present value of the regular monthly payments is approximately \$16,152. Therefore, the remaining present value to be covered by the special payment $ X $ is approximately \$35,785.

Next, we will use the equation for the present value of the special payment:
\[
PV_{\text{X}} = \frac{X}{(1 + r)^{10}}
\]
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 guide you through the remaining steps for solving the equation manually:

1. Use the equation for the present value of the special payment:
   \[
  

Solve for the special payment X in an investment problem with regular monthly payments of $1,396, a special payment in 10 months, and an expected return of 0.67% per month. This problem uses the Present Value of Annuity and Lump-Sum Payment formulas, suitable for advanced high school or college-level students.

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