If a,b,c be the pth , qth and rth terms of an A.P., then `p(b-c) + q(c-a) + r(a-b)` equals to :

Options

A. 0
B. 1
C. 3
D. `(abc)/(pqr)`

Answer

To solve the problem, we need to evaluate the expression \( p(b-c) + q(c-a) + r(a-b) \) given that \( a, b, c \) are the \( p \)-th, \( q \)-th, and \( r \)-th terms of an Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Identify the terms of the A.P.**:
- The \( p \)-th term \( a \) can be expressed as:
\[ a = A + (p-1)d \]
- The \( q \)-th term \( b \) can be expressed as:
\[ b = A + (q-1)d \]
- The \( r \)-th term \( c \) can be expressed as:
\[ c = A + (r-1)d \]
where \( A \) is the first term and \( d \) is the common difference.

2. **Substitute the expressions for \( a, b, c \) into the expression**:
\[ p(b-c) + q(c-a) + r(a-b) \]

3. **Calculate \( b - c \)**:
\[ b - c = \left(A + (q-1)d\right) - \left(A + (r-1)d\right) = (q - r)d \]

4. **Calculate \( c - a \)**:
\[ c - a = \left(A + (r-1)d\right) - \left(A + (p-1)d\right) = (r - p)d \]

5. **Calculate \( a - b \)**:
\[ a - b = \left(A + (p-1)d\right) - \left(A + (q-1)d\right) = (p - q)d \]

6. **Substitute these differences back into the expression**:
\[ p(b-c) = p(q - r)d \]
\[ q(c-a) = q(r - p)d \]
\[ r(a-b) = r(p - q)d \]

7. **Combine all parts**:
\[ p(q - r)d + q(r - p)d + r(p - q)d \]

8. **Factor out \( d \)**:
\[ = d \left[ p(q - r) + q(r - p) + r(p - q) \right] \]

9. **Simplify the expression inside the brackets**:
- Expanding:
\[ = pq - pr + qr - qp + rp - rq \]
- Rearranging:
\[ = (pq - qp) + (qr - rq) + (rp - pr) = 0 \]

10. **Final Result**:
\[ p(b-c) + q(c-a) + r(a-b) = 0 \]

### Conclusion:
The value of the expression \( p(b-c) + q(c-a) + r(a-b) \) is \( 0 \).