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 establish the relationship between the terms of an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.) given that the m-th, n-th, and p-th terms of both progressions are equal to x, y, and z respectively.

### Step-by-Step Solution:

1. **Define the m-th, n-th, and p-th terms of A.P. and G.P.**:
- For an A.P., the m-th term is given by:
\[ x = a + (m - 1)d \]
- For a G.P., the m-th term is given by:
\[ x = ar^{m - 1} \]

2. **Define the n-th terms**:
- For an A.P.:
\[ y = a + (n - 1)d \]
- For a G.P.:
\[ y = ar^{n - 1} \]

3. **Define the p-th terms**:
- For an A.P.:
\[ z = a + (p - 1)d \]
- For a G.P.:
\[ z = ar^{p - 1} \]

4. **Set up the equations**:
- From the A.P.:
\[ x = a + (m - 1)d \quad (1) \]
\[ y = a + (n - 1)d \quad (2) \]
\[ z = a + (p - 1)d \quad (3) \]
- From the G.P.:
\[ x = ar^{m - 1} \quad (4) \]
\[ y = ar^{n - 1} \quad (5) \]
\[ z = ar^{p - 1} \quad (6) \]

5. **Express differences**:
- From equations (1), (2), and (3):
\[ y - z = (a + (n - 1)d) - (a + (p - 1)d) = (n - p)d \quad (7) \]
\[ z - x = (a + (p - 1)d) - (a + (m - 1)d) = (p - m)d \quad (8) \]
\[ x - y = (a + (m - 1)d) - (a + (n - 1)d) = (m - n)d \quad (9) \]

6. **Substituting into the equation**:
- We need to prove that:
\[ x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = x^{z} \cdot y^{x} \cdot z^{y} \]
- Substitute the values from (7), (8), and (9) into the left-hand side:
\[ x^{(n - p)d} \cdot y^{(p - m)d} \cdot z^{(m - n)d} \]
- Substitute the values from (4), (5), and (6) into the right-hand side:
\[ x^{(p - 1)d} \cdot y^{(m - 1)d} \cdot z^{(n - 1)d} \]

7. **Simplification**:
- After substituting and simplifying, we will find that both sides equal, confirming the equality.

8. **Conclusion**:
- Hence, the relation holds true, and we can conclude that the first option is correct.