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To solve for the values of \( X \) and \( Y \) so that the three cash flow series are equivalent at an interest rate of 15% per year (compounded annually), we need to equate the present value (PV) of each cash flow series.

The present value of a cash flow series can be calculated using the formula:

\[
PV = \sum_{t=0}^{n} \frac{C_t}{(1 + r)^t}
\]

Where:
- \( C_t \) is the cash flow at time \( t \),
- \( r \) is the interest rate (15% or 0.15 in this case),
- \( t \) is the time period (in years),
- \( n \) is the number of years (here, \( n = 5 \)).

We will calculate the present value of each series (A, B, and C) and equate them to find \( X \) and \( Y \).

### Cash Flow Series A:
- Year 0: \( -1,000 \)
- Year 1: \( X \)
- Year 2: \( 1.5X \)
- Year 3: \( 2.0X \)
- Year 4: \( 2.5X \)
- Year 5: \( 3.0X \)

The present value of Series A is:

\[
PV_A = \frac{-1,000}{(1 + 0.15)^0} + \frac{X}{(1 + 0.15)^1} + \frac{1.5X}{(1 + 0.15)^2} + \frac{2.0X}{(1 + 0.15)^3} + \frac{2.5X}{(1 + 0.15)^4} + \frac{3.0X}{(1 + 0.15)^5}
\]

### Cash Flow Series B:
- Year 0: \( -2,500 \)
- Year 1: \( 3,000 \)
- Year 2: \( 2,500 \)
- Year 3: \( 2,000 \)
- Year 4: \( 1,500 \)
- Year 5: \( 1,000 \)

The present value of Series B is:

\[
PV_B = \frac{-2,500}{(1 + 0.15)^0} + \frac{3,000}{(1 + 0.15)^1} + \frac{2,500}{(1 + 0.15)^2} + \frac{2,000}{(1 + 0.15)^3} + \frac{1,500}{(1 + 0.15)^4} + \frac{1,000}{(1 + 0.15)^5}
\]

### Cash Flow Series C:
- Year 0: \( 0 \)
- Year 1: \( Y \)
- Year 2: \( Y \)
- Year 3: \( Y \)
- Year 4: \( 2Y \)
- Year 5: \( 2Y \)

The present value of Series C is:

\[
PV_C = \frac{Y}{(1 + 0.15)^1} + \frac{Y}{(1 + 0.15)^2} + \frac{Y}{(1 + 0.15)^3} + \frac{2Y}{(1 + 0.15)^4} + \frac{2Y}{(1 + 0.15)^5}
\]

We now set the present values of Series A, B, and C equal to each other:

\[
PV_A = PV_B = PV_C
\]

I'll calculate the individual present values for Series B, and then solve for \( X \) and \( Y \) by equating them. Let me calculate that.The present value of Cash Flow Series B is approximately \( 4,668.89 \).

Next, I'll set the present value of Series A and Series C equal to this amount, and solve for \( X \) and \( Y \).

Let me compute the present value equations for Series A and C now.The values of \( X \) and \( Y \) that make all three cash flow series equivalent at a 15% interest rate are:

\[
X \approx 908.52
\]
\[
Y \approx 1056.05
\]

Would you like further details or have any other questions?

Here are five related questions you might explore:
1. How would the solution change if the interest rate was compounded semi-annually?
2. What is the effect on present value if the cash flow occurs at the beginning of the year rather than the end?
3. How do these cash flows compare to an annuity with fixed payments over the same period?
4. What is the future value of these cash flows after 5 years at the same interest rate?
5. How would you interpret the net present value of a project using these cash flows?

**Tip**: When solving cash flow problems with different interest rates, always check if the compounding period matches the cash flow period to ensure accuracy.

Determine the values of X and Y so that all three cash flow series are equivalent at an interest rate of 15% per year compounded yearly, given the following cash flow series for the next 5 years.

Learn how to determine the values of X and Y for three cash flow series to be equivalent at a 15% interest rate compounded yearly. This problem involves calculating the present value of each series using the formula PV = Σ(C_t / (1 + r)^t) and equating them. Suitable for advanced high school or college students.

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