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 find the \( p^{th} \) term of the arithmetic progression (A.P.), given that the \( m^{th} \) term is \( n \) and the \( n^{th} \) term is \( m \), we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the A.P. Terms**:
- The \( m^{th} \) term of an A.P. can be expressed as:
\[ A_m = A + (m - 1)D \]
where \( A \) is the first term and \( D \) is the common difference.
- According to the problem, we have:
\[ A_m = n \quad \text{(1)} \]

2. **Expressing the \( n^{th} \) Term**:
- Similarly, the \( n^{th} \) term can be expressed as:
\[ A_n = A + (n - 1)D \]
and we know:
\[ A_n = m \quad \text{(2)} \]

3. **Setting Up the Equations**:
- From equation (1):
\[ A + (m - 1)D = n \]
- From equation (2):
\[ A + (n - 1)D = m \]

4. **Subtracting the Two Equations**:
- Subtract equation (1) from equation (2):
\[ [A + (n - 1)D] - [A + (m - 1)D] = m - n \]
- This simplifies to:
\[ (n - 1)D - (m - 1)D = m - n \]
- Rearranging gives:
\[ (n - m)D = m - n \]
- Therefore:
\[ D(n - m) = -(n - m) \]
- If \( n \neq m \), we can divide both sides by \( n - m \):
\[ D = -1 \]

5. **Finding the Value of \( A \)**:
- Substitute \( D = -1 \) back into either equation (1) or (2). Using equation (1):
\[ A + (m - 1)(-1) = n \]
- This simplifies to:
\[ A - m + 1 = n \]
- Rearranging gives:
\[ A = n + m - 1 \quad \text{(3)} \]

6. **Finding the \( p^{th} \) Term**:
- The \( p^{th} \) term of the A.P. is given by:
\[ A_p = A + (p - 1)D \]
- Substituting \( A \) from equation (3) and \( D = -1 \):
\[ A_p = (n + m - 1) + (p - 1)(-1) \]
- This simplifies to:
\[ A_p = n + m - 1 - p + 1 \]
- Therefore:
\[ A_p = n + m - p \]

### Final Answer:
The \( p^{th} \) term of the A.P. is:
\[ \boxed{n + m - p} \]