Question

If $x>1,y>1,z>1$ are in GP, then $\frac{1}{1+lnx},\frac{1}{1+lny},\frac{1}{1+lnz}$ are in

• A. AP
• B. HP
• C. GP
• D. None of these

Answer 1

The correct option is B HP

Let the common ratio of the GP be $r$

Then $y=xr$ and $z=x{r}^2$.
$\Rightarrow lny=lnr+lnr$ and $lnz=lnx+2lnr$
Putting $A=1_{l}nx,D=lnr$
Then, $\frac{1}{1+lnr}=\frac{1}{A}$,
$\frac{1}{1+lny}=\frac{1}{1+lnxr}=\frac{1}{lnr+lnr}=\frac{1}{A+D}$
and $\frac{1}{1+lnz}=\frac{1}{1+lnx+2lnr}=\frac{1}{A+2D}$
Here we see that $A,A+D$ and $A+2D$ are in AP.
Therefore, $\frac{1}{A},\frac{1}{A+D}$ and $\frac{1}{A+2D}$ are in HP.
Therefore, $\frac{1}{1+lnx},\frac{1}{1+lny},\frac{1}{1+lnz}$, are in HP.