If a,b,c be the pth , qth, rth terms of a GP then the value of (q-r)log a+ (r-p) log b + (p-q) log c is:

Options

A. 0
B. 1
C. `-1`
D. pqr

Answer

To solve the problem, we need to find the value of the expression \((q - r) \log a + (r - p) \log b + (p - q) \log c\) given that \(a\), \(b\), and \(c\) are the \(p\)th, \(q\)th, and \(r\)th terms of a geometric progression (GP).

### Step-by-Step Solution:

1. **Understanding the Terms of GP**:
- In a geometric progression, the \(n\)th term can be expressed as:
\[ T_n = A \cdot r^{n-1} \]
where \(A\) is the first term and \(r\) is the common ratio.

2. **Expressing the Terms**:
- The \(p\)th term \(a\) can be written as:
\[ a = A \cdot r^{p-1} \]
- The \(q\)th term \(b\) can be expressed as:
\[ b = A \cdot r^{q-1} \]
- The \(r\)th term \(c\) can be expressed as:
\[ c = A \cdot r^{r-1} \]

3. **Substituting the Terms into the Expression**:
- Now, substitute \(a\), \(b\), and \(c\) into the expression:
\[ (q - r) \log a + (r - p) \log b + (p - q) \log c \]
- This becomes:
\[ (q - r) \log(A \cdot r^{p-1}) + (r - p) \log(A \cdot r^{q-1}) + (p - q) \log(A \cdot r^{r-1}) \]

4. **Using Logarithm Properties**:
- Using the property \(\log(xy) = \log x + \log y\), we can expand each term:
\[ (q - r) [\log A + (p - 1) \log r] + (r - p) [\log A + (q - 1) \log r] + (p - q) [\log A + (r - 1) \log r] \]

5. **Combining the Terms**:
- Combine the logarithmic terms:
\[ (q - r) \log A + (r - p) \log A + (p - q) \log A + (q - r)(p - 1) \log r + (r - p)(q - 1) \log r + (p - q)(r - 1) \log r \]
- The first part simplifies to:
\[ [(q - r) + (r - p) + (p - q)] \log A = 0 \cdot \log A = 0 \]

6. **Simplifying the Second Part**:
- Now, focus on the second part:
\[ (q - r)(p - 1) \log r + (r - p)(q - 1) \log r + (p - q)(r - 1) \log r \]
- This can be factored out as:
\[ \log r \left[(q - r)(p - 1) + (r - p)(q - 1) + (p - q)(r - 1)\right] \]
- The expression inside the brackets simplifies to zero, as it represents a cyclic sum of differences.

7. **Final Result**:
- Therefore, the entire expression simplifies to:
\[ 0 \]

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