If x,y , z are in A.P then `(x-y)/(y-z)` is equal to _________-

Options

A. `½`
B. 2
C. 1
D. 0

Answer

To solve the problem, we need to find the value of \((x - y) / (y - z)\) given that \(x\), \(y\), and \(z\) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P.**:
- If \(x\), \(y\), and \(z\) are in A.P., then by definition, the middle term is the average of the other two terms. This can be mathematically expressed as:
\[ 2y = x + z \]

2. **Rearranging the A.P. Equation**:
- From the equation \(2y = x + z\), we can express \(x\) in terms of \(y\) and \(z\):
\[ x = 2y - z \]

3. **Substituting in the Expression**:
- Now, we need to substitute \(x\) in the expression \((x - y) / (y - z)\):
\[ \frac{x - y}{y - z} = \frac{(2y - z) - y}{y - z} \]

4. **Simplifying the Numerator**:
- Simplifying the numerator:
\[ (2y - z) - y = 2y - y - z = y - z \]

5. **Final Expression**:
- Now we can rewrite the expression:
\[ \frac{y - z}{y - z} \]
- Since \(y - z\) in the numerator and denominator are the same, they cancel out:
\[ = 1 \]

### Conclusion:
Thus, if \(x\), \(y\), and \(z\) are in A.P., then:
\[ \frac{x - y}{y - z} = 1 \]