If the `m^(th),n^(th)andp^(th)` terms of an A.P. and G.P. be equal and be respectively x,y,z, then

Options

A. `x^(y)y^(z)z^(x)=x^(z)y^(x)z^(y)`
B. `(x-y)^(x)(y-z)^(x)=(z-x)^(z)`
C. `(x-y)^(z)(y-z)^(x)=(z-x)^(y)`
D. none of these

Answer

To solve the problem, we need to find the relationship between the terms \( x, y, z \) given that the \( m^{th}, n^{th}, \) and \( p^{th} \) terms of an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.) are equal to \( x, y, z \) respectively.

### Step-by-step Solution:

1. **Identify the terms in A.P.**:
- The \( m^{th} \) term of an A.P. is given by:
\[ x = a + (m - 1)d \]
- The \( n^{th} \) term of an A.P. is given by:
\[ y = a + (n - 1)d \]
- The \( p^{th} \) term of an A.P. is given by:
\[ z = a + (p - 1)d \]

2. **Identify the terms in G.P.**:
- The \( m^{th} \) term of a G.P. is given by:
\[ x = A \cdot r^{m - 1} \]
- The \( n^{th} \) term of a G.P. is given by:
\[ y = A \cdot r^{n - 1} \]
- The \( p^{th} \) term of a G.P. is given by:
\[ z = A \cdot r^{p - 1} \]

3. **Set up equations**:
- From the A.P. terms:
\[ x - y = (m - n)d \quad (1) \]
\[ y - z = (n - p)d \quad (2) \]
\[ z - x = (p - m)d \quad (3) \]

4. **Set up equations from G.P. terms**:
- From the G.P. terms:
\[ x - y = A(r^{m - 1} - r^{n - 1}) = A r^{n - 1}(r^{m - n} - 1) \quad (4) \]
\[ y - z = A(r^{n - 1} - r^{p - 1}) = A r^{p - 1}(r^{n - p} - 1) \quad (5) \]
\[ z - x = A(r^{p - 1} - r^{m - 1}) = A r^{m - 1}(r^{p - m} - 1) \quad (6) \]

5. **Equate the differences**:
- From equations (1) and (4):
\[ (m - n)d = A r^{n - 1}(r^{m - n} - 1) \quad (7) \]
- From equations (2) and (5):
\[ (n - p)d = A r^{p - 1}(r^{n - p} - 1) \quad (8) \]
- From equations (3) and (6):
\[ (p - m)d = A r^{m - 1}(r^{p - m} - 1) \quad (9) \]

6. **Combine the results**:
- We can combine the results from equations (7), (8), and (9) to derive a relationship between \( x, y, z \):
\[ \frac{x - y}{y - z} \cdot \frac{y - z}{z - x} \cdot \frac{z - x}{x - y} = 1 \]
- This leads us to the conclusion:
\[ x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = 1 \]

### Final Result:
Thus, the required relation is:
\[ x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = 1 \]