Question

If $x > 1, y > 1, z > 1$ are in $G.P.$ then $\frac{1}{1+log\,x}, \frac{1}{1+log\,y}, \frac{1}{1+log\,z}$ are in :

• A. $A.P$
• B. $H.P$
• C. $G.P$
• D. None of these

Answer 1

Answer: (B)

$H.P$

Answer 2

If $x > 1, y > 1, z > 1$ are in $G.P.$ $\therefore y^{2}=xz\,...\left(i\right)$ Taking log on both sides of equ. $\left(i\right)$, we get $2\,log\,y=log\,x+log\,z$ $\Rightarrow 2+2\,log\,y=2+log\,x+log\,z$ $\Rightarrow 2\left(1+log\,y\right)=\left(1+log\,x\right)+\left(1+log\,z\right)$ Clearly, $\left(1 + log \,x\right), \left(1 + log \,y\right) \left(1 + log \,z\right)$ are in $A.P.$ Then $\frac{1}{1+log\,x}, \frac{1}{1+log\,y}, \frac{1}{log\,z}$ are in $H.P.$