Question

If $a,b,c$ are respectively the $x^{th},y^{th}$ and $z^{th}$ terms of a G.P., then $(y-z)\log a+(z-x)\log b+(x-y)\log c=$

• A. $1$
• B. $2$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

Let $A$ be the first term and $R$, the common ratio of G.P.

Then $a=t_x=A{R}^{x-1},b=t_y=A{R}^{y-1}$ and $c=t_z=A{R}^{z-1}$

$\therefore (y-z)\log a+(z-x)\log b+(x-y)\log c=\log a^{y-z}+\log b^{z-x}+\log c^{x-y}=\log (a^{y-z}.b^{z-x}.c^{x-y})=\log [{\left(A{R}^{x-1}\right)}^{y-z}.{\left(A{R}^{y-1}\right)}^{z-x}.{\left(A{R}^{z-1}\right)}^{x-y}]=\log [A^{y-z+z-x+x-y}.R^{(x-1)(y-z)+(y-1)(z-x)+(z-1)(x-y)}]=\log (A^0\times R^0)=\log 1=0$