Question

If 2 $log(x-z)-log(x-2y+z)=log(x+z)$, then $x,y$ and $z$ are in

• A. A.P
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

The correct option is C H.P

Given

$2\log (x-z)-\log (z-2y+z)=\log (x+z)$

$\therefore 2\log (x-z)=\log (x-2y+z)+\log (x+z)$

$\therefore \log \left[\right(x-z{)}^2]=\log [(x-2y+z)(x+z)]$

$\therefore (x-z{)}^2=(x+z\left)\right((x+z)-2y)$

$(x-z{)}^2=(x+z{)}^2-2y(x+z)$

$\therefore 2y(x+z)=(x+z{)}^2-(x-z{)}^2$

$2y(x+z)=(x+z+x-z)(x+z-x+z)$

$=2y(x+z)=4xz$

$\therefore \frac{1}{x}+\frac{1}{z}=\frac{2}{z}$

$\Rightarrow x,y,z$ is HP.