To solve the problem, we need to analyze the information given about the two arithmetic progressions (APs) \( x_1, x_2, x_3, x_4, x_5 \) and \( y_1, y_2, y_3, y_4, y_5 \).
### Step 1: Define the terms of the APs
Let the first arithmetic progression be defined as:
- \( x_1 = a - 2d \)
- \( x_2 = a - d \)
- \( x_3 = a \)
- \( x_4 = a + d \)
- \( x_5 = a + 2d \)
Similarly, for the second arithmetic progression:
- \( y_1 = b - 2e \)
- \( y_2 = b - e \)
- \( y_3 = b \)
- \( y_4 = b + e \)
- \( y_5 = b + 2e \)
### Step 2: Use the sum condition
We know that:
\[
\sum_{i=1}^{5} x_i = \sum_{i=1}^{5} y_i = 25
\]
Calculating the sum of the \( x_i \):
\[
x_1 + x_2 + x_3 + x_4 + x_5 = (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 5a
\]
Thus, we have:
\[
5a = 25 \implies a = 5
\]
Now calculating the sum of the \( y_i \):
\[
y_1 + y_2 + y_3 + y_4 + y_5 = (b - 2e) + (b - e) + b + (b + e) + (b + 2e) = 5b
\]
Thus, we have:
\[
5b = 25 \implies b = 5
\]
### Step 3: Use the product condition
We also know that:
\[
\prod_{i=1}^{5} x_i = \prod_{i=1}^{5} y_i = 0
\]
Since the product of the \( x_i \) is zero, at least one of the \( x_i \) must be zero. We can find the roots of the polynomial formed by the \( x_i \):
\[
x_1 = 5 - 2d, \quad x_2 = 5 - d, \quad x_3 = 5, \quad x_4 = 5 + d, \quad x_5 = 5 + 2d
\]
Setting \( x_3 = 5 = 0 \) gives us \( d = \frac{5}{2} \) or \( d = 5 \).
### Step 4: Calculate the values of \( x_i \) and \( y_i \)
1. If \( d = \frac{5}{2} \):
- \( x_1 = 0 \)
- \( x_2 = \frac{5}{2} \)
- \( x_3 = 5 \)
- \( x_4 = \frac{15}{2} \)
- \( x_5 = 10 \)
2. If \( d = 5 \):
- \( x_1 = -5 \)
- \( x_2 = 0 \)
- \( x_3 = 5 \)
- \( x_4 = 10 \)
- \( x_5 = 15 \)
### Step 5: Calculate \( y_i \) values
For both cases, we can find \( y_i \) similarly using \( e \):
1. If \( e = \frac{5}{2} \):
- \( y_1 = -5 \)
- \( y_2 = 0 \)
- \( y_3 = 5 \)
- \( y_4 = 10 \)
- \( y_5 = 15 \)
2. If \( e = 5 \):
- \( y_1 = 0 \)
- \( y_2 = 5 \)
- \( y_3 = 10 \)
- \( y_4 = 15 \)
- \( y_5 = 20 \)
### Step 6: Calculate \( |y_5 - x_5| \)
For the first case:
\[
|y_5 - x_5| = |15 - 10| = 5
\]
For the second case:
\[
|y_5 - x_5| = |20 - 15| = 5
\]
### Conclusion
In both scenarios, we find that:
\[
|y_5 - x_5| = 5
\]
Thus, the answer is \( \boxed{5} \).
