The ` m^(th)` term of an A.P. is n and `n^(th)` term is m its
` p^(th)` term is

Options

A. `m-n+p`
B. `n+p -m`
C. `m+n-p`
D. `m+n+p`

Answer

To solve the problem, we need to find the \( p^{th} \) term of an arithmetic progression (A.P.) given that the \( m^{th} \) term is \( n \) and the \( n^{th} \) term is \( m \).

### Step-by-Step Solution:

1. **Understanding the A.P. Terms**:
The \( m^{th} \) term of an A.P. can be expressed as:
\[ a + (m - 1)d = n \quad \text{(Equation 1)} \]
The \( n^{th} \) term of the A.P. can be expressed as:
\[ a + (n - 1)d = m \quad \text{(Equation 2)} \]

2. **Subtracting the Two Equations**:
To eliminate \( a \), we subtract Equation 2 from Equation 1:
\[ (a + (m - 1)d) - (a + (n - 1)d) = n - m \]
This simplifies to:
\[ (m - 1)d - (n - 1)d = n - m \]
Which further simplifies to:
\[ (m - n)d = n - m \]

3. **Solving for \( d \)**:
Rearranging the equation gives:
\[ (m - n)d = -(m - n) \]
If \( m \neq n \), we can divide both sides by \( (m - n) \):
\[ d = -1 \]

4. **Finding \( a \)**:
Substitute \( d = -1 \) back into Equation 1 to find \( a \):
\[ a + (m - 1)(-1) = n \]
This simplifies to:
\[ a - (m - 1) = n \]
Therefore:
\[ a = n + m - 1 \]

5. **Finding the \( p^{th} \) Term**:
The \( p^{th} \) term of the A.P. is given by:
\[ a + (p - 1)d \]
Substituting the values of \( a \) and \( d \):
\[ (n + m - 1) + (p - 1)(-1) \]
This simplifies to:
\[ n + m - 1 - (p - 1) \]
Which further simplifies to:
\[ n + m - p \]

6. **Final Result**:
Therefore, the \( p^{th} \) term of the A.P. is:
\[ m + n - p \]

### Final Answer:
The \( p^{th} \) term is \( m + n - p \).