First slide

Geometric progression

Question

If a, b, c are respectively the xth, yth and zth terms of a G.P., then $(y - z) \log a + (z - x) \log b + (x - y) \log c =$

Moderate

A

1
B

-1
C

0
D

None of these

Solution

Let A be the first term and R, the common ratio of G.P.
Then $a = a_{x} = {AR}^{x - 1}, b = a_{v} = {AR}^{y - 1}$
and $c = a_{z} = {AR}^{z - 1}$
$\begin{array}{l} ∴ (y - z) \log a + (z - x) \log b + (x - y) \log c \\ = \log a^{y - z} + \log b^{z - x} + \log c^{x - y} \\ = \log (a^{y - z} \cdot b^{z - x} \cdot c^{x - y}) \\ = \log [{({AR}^{x - 1})}^{y - z} \times {({AR}^{y - 1})}^{z - x} \cdot {({AR}^{z - 1})}^{x - y}] \\ = \log [A_{(x - y)}^{y - z + z - x + x - y} \cdot R^{(x - 1) (y - z) + (y - 1) (z - x) + (z - 1)} \\ = \log (A^{0} \times R^{0}) = \log 1 = 0 . \end{array}$

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