If the pth, qth and rth terms of an A.P. are a,b,c respectively , then the value of `a(q-r) + b(r-p) + c(p-q) ` is :

Options

A. 0
B. 1
C. abc
D. pqr

Answer

To solve the problem, we need to find the value of the expression \( a(q - r) + b(r - p) + c(p - q) \), given that the \( p \)-th, \( q \)-th, and \( r \)-th terms of an arithmetic progression (A.P.) are \( a \), \( b \), and \( c \) respectively.

### Step-by-Step Solution:

1. **Identify the General Formula for the Terms of an A.P.**:
The \( n \)-th term of an A.P. can be expressed as:
\[ T_n = A + (n - 1)D \]
where \( A \) is the first term and \( D \) is the common difference.

2. **Express the Given Terms**:
- The \( p \)-th term is:
\[ T_p = A + (p - 1)D = a \]
- The \( q \)-th term is:
\[ T_q = A + (q - 1)D = b \]
- The \( r \)-th term is:
\[ T_r = A + (r - 1)D = c \]

3. **Rearranging the Equations**:
From the above equations, we can express \( A \) in terms of \( a \), \( b \), and \( c \):
- From \( T_p \):
\[ A = a - (p - 1)D \]
- From \( T_q \):
\[ A = b - (q - 1)D \]
- From \( T_r \):
\[ A = c - (r - 1)D \]

4. **Setting Up the Expression**:
We need to evaluate:
\[ a(q - r) + b(r - p) + c(p - q) \]

5. **Substituting for \( a \), \( b \), and \( c \)**:
Substitute \( a \), \( b \), and \( c \) from the rearranged equations into the expression:
\[ = (A + (p - 1)D)(q - r) + (A + (q - 1)D)(r - p) + (A + (r - 1)D)(p - q) \]

6. **Expanding the Expression**:
Expanding each term:
\[ = A(q - r) + (p - 1)D(q - r) + A(r - p) + (q - 1)D(r - p) + A(p - q) + (r - 1)D(p - q) \]

7. **Combining Like Terms**:
Combine all terms involving \( A \) and \( D \):
\[ = A[(q - r) + (r - p) + (p - q)] + D[(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)] \]

8. **Simplifying the Coefficients of \( A \)**:
Notice that:
\[ (q - r) + (r - p) + (p - q) = 0 \]
Therefore, the first part becomes \( 0 \).

9. **Simplifying the Coefficients of \( D \)**:
The second part can also be simplified:
\[ (p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q) = 0 \]
This can be shown by expanding and rearranging terms, leading to cancellation.

10. **Final Result**:
Thus, the entire expression simplifies to:
\[ 0 \]

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