If p, q, r are in A.P. and x, y, z in G.P., then `x^(q-r) y^(r-p) z^(p - q) =`

Options

A. -1
B. 1
C. 2
D. none

Answer

To solve the problem, we need to find the value of \( x^{(q-r)} y^{(r-p)} z^{(p-q)} \) given that \( p, q, r \) are in Arithmetic Progression (A.P.) and \( x, y, z \) are in Geometric Progression (G.P.).

### Step-by-Step Solution:

1. **Understanding A.P.**:
Since \( p, q, r \) are in A.P., we can express the relationships between them using a common difference \( d \):
\[ q - p = d \quad \text{and} \quad r - q = d \]
From these, we can derive:
\[ r - p = (r - q) + (q - p) = d + d = 2d \]

2. **Expressing Differences**:
We can now express the differences we need:
- \( q - r = -d \)
- \( r - p = 2d \)
- \( p - q = -d \)

3. **Understanding G.P.**:
Since \( x, y, z \) are in G.P., we have the relationship:
\[ y^2 = xz \]

4. **Substituting the Differences**:
Now we substitute these values into the expression \( x^{(q-r)} y^{(r-p)} z^{(p-q)} \):
\[ x^{(q-r)} = x^{-d}, \quad y^{(r-p)} = y^{2d}, \quad z^{(p-q)} = z^{-d} \]
Therefore, the expression becomes:
\[ x^{-d} y^{2d} z^{-d} \]

5. **Rearranging the Expression**:
We can rearrange this as:
\[ \frac{y^{2d}}{x^d z^d} \]

6. **Using the G.P. Relationship**:
From the G.P. relationship \( y^2 = xz \), we can substitute \( y^2 \) in our expression:
\[ \frac{y^{2d}}{x^d z^d} = \frac{(xz)^d}{x^d z^d} \]

7. **Simplifying**:
This simplifies to:
\[ \frac{(xz)^d}{x^d z^d} = \frac{x^d z^d}{x^d z^d} = 1 \]

### Final Answer:
Thus, we conclude that:
\[ x^{(q-r)} y^{(r-p)} z^{(p-q)} = 1 \]