If the pth term of an A.P. is q and its qth term is p then its mth term is :

Options

A. `(pq)/(m)`
B. pq-m
C. p+q-m
D. pqm

Answer

To find the mth term of the arithmetic progression (A.P.) given that the pth term is q and the qth term is p, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Set up the equations based on the given information**:
- For the pth term:
\[ T_p = a + (p - 1) \cdot d = q \quad \text{(1)} \]
- For the qth term:
\[ T_q = a + (q - 1) \cdot d = p \quad \text{(2)} \]

3. **Rearrange both equations**:
- From equation (1):
\[ a + (p - 1) \cdot d = q \implies a = q - (p - 1) \cdot d \quad \text{(3)} \]
- From equation (2):
\[ a + (q - 1) \cdot d = p \implies a = p - (q - 1) \cdot d \quad \text{(4)} \]

4. **Equate equations (3) and (4)**:
\[ q - (p - 1) \cdot d = p - (q - 1) \cdot d \]
Rearranging gives:
\[ q - p = (p - 1) \cdot d - (q - 1) \cdot d \]
Simplifying:
\[ q - p = (p - q) \cdot d \]

5. **Solve for d**:
\[ d = \frac{q - p}{p - q} = -1 \quad \text{(since \( p - q \) is negative)} \]

6. **Substitute d back to find a**:
Substitute \( d = -1 \) into equation (3):
\[ a = q - (p - 1)(-1) = q + p - 1 \]

7. **Find the mth term**:
Using the formula for the mth term:
\[ T_m = a + (m - 1) \cdot d \]
Substitute \( a \) and \( d \):
\[ T_m = (q + p - 1) + (m - 1)(-1) \]
Simplifying:
\[ T_m = q + p - 1 - m + 1 = p + q - m \]

### Final Result:
Thus, the mth term of the A.P. is:
\[ T_m = p + q - m \]