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 involves finding the special payment $ X $ in a time value of money scenario, where you have a series of monthly payments, a special payment in month 10, and a given expected return. We will use the concept of the Present Value (PV) of an annuity with an additional lump sum payment to calculate $ X $.

### Key Details:
- Investment value (Present Value, $ PV $): $ \$51,937 $
- Monthly payments: $ \$1,396 $
- Duration of monthly payments: 12 months
- Special payment $ X $ in month 10
- Expected return: $ 0.67\% $ per month
- The first monthly payment is made today (implies it is an annuity due).

#### Step-by-step Solution:

1. **Identify the formula for Present Value (PV) of an annuity due**:
   
   For regular payments starting today, the PV of an annuity due is given by:
   \[
   PV_{\text{annuity}} = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
   \]
   where:
   - $ C = 1,396 $ (monthly payment),
   - $ r = 0.0067 $ (monthly interest rate),
   - $ n = 12 $ (total number of monthly payments).

2. **Calculate the PV of the regular payments**.

3. **Identify the formula for Present Value of the lump sum payment $ X $**:
   
   The PV of the lump sum $ X $ paid at month 10 is:
   \[
   PV_X = \frac{X}{(1 + r)^{10}}
   \]

4. **Sum the PV of the regular payments and the PV of $ X $** and set it equal to the investment value $ \$51,937 $.

I will now calculate $ X $.The special payment $ X $ that needs to be made in 10 months is approximately **\$38,256**.

Would you like further details on this calculation, or have any other questions?

Here are 5 follow-up questions:
1. How is the present value of an annuity different from the future value?
2. What is the significance of using an annuity due in this problem?
3. How would the result change if the special payment occurred in month 12 instead of month 10?
4. What would happen to the special payment $ X $ if the monthly return rate was higher?
5. Can this method be used for irregular payments over time?

**Tip:** In an annuity due, the first payment is made at the start of the period, which slightly increases the present value compared to an ordinary annuity where payments are made at the end.

Learn how to calculate the special payment X for an investment with regular monthly payments of $1,396 and a special payment in month 10, given a 0.67% monthly return. This problem covers key concepts in the time value of money, including the present value of an annuity due and lump sum payment.

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