If a,b,c are respectively the xth, yth and zth terms of a G.P. then the value of
`(y-z)log a + (z-x)log b+(x-y) logc`:

Options

A. 0
B. 1
C. 3
D. none of these

Answer

To solve the problem, we need to find the value of the expression:

\[ (y-z) \log a + (z-x) \log b + (x-y) \log c \]

where \(a\), \(b\), and \(c\) are the \(x\)th, \(y\)th, and \(z\)th terms of a geometric progression (G.P.).

### Step 1: Understand the terms in G.P.

In a geometric progression, the \(n\)th term can be expressed as:

\[ T_n = ar^{n-1} \]

where \(a\) is the first term and \(r\) is the common ratio. Thus, we can express the terms \(a\), \(b\), and \(c\) as:

\[ a = T_x = ar^{x-1} \]
\[ b = T_y = ar^{y-1} \]
\[ c = T_z = ar^{z-1} \]

### Step 2: Substitute the terms into the expression

Now, substituting the values of \(a\), \(b\), and \(c\) into the expression, we get:

\[ (y-z) \log(ar^{x-1}) + (z-x) \log(ar^{y-1}) + (x-y) \log(ar^{z-1}) \]

### Step 3: Apply the logarithm properties

Using the property of logarithms, \(\log(mn) = \log m + \log n\), we can expand each term:

\[ = (y-z)(\log a + (x-1) \log r) + (z-x)(\log a + (y-1) \log r) + (x-y)(\log a + (z-1) \log r) \]

### Step 4: Distribute and combine like terms

Distributing each term, we have:

\[ = (y-z) \log a + (y-z)(x-1) \log r + (z-x) \log a + (z-x)(y-1) \log r + (x-y) \log a + (x-y)(z-1) \log r \]

Now, combine the terms involving \(\log a\):

\[ = [(y-z) + (z-x) + (x-y)] \log a + [(y-z)(x-1) + (z-x)(y-1) + (x-y)(z-1)] \log r \]

### Step 5: Simplify the coefficients of \(\log a\)

Notice that the coefficients of \(\log a\) simplify to zero:

\[ (y-z) + (z-x) + (x-y) = 0 \]

### Step 6: Conclude the expression

Thus, the entire expression simplifies to:

\[ 0 \cdot \log a + \text{(some expression)} \cdot \log r = 0 \]

Therefore, the value of the original expression is:

\[ \boxed{0} \]