Question

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

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

Answer 1

Answer: (B)

Let the common ratio of the GP be $r$.
Then, $y=xr$ and $z=x{r}^2$
$\Rightarrow \ln y=\ln x+\ln r$ and $\ln z=\ln x+2\ln r$
Putting $A=1+\ln x,D=\ln r$
Then, $\frac{1}{1+\ln x}=\frac{1}{A}$
$\frac{1}{1+\ln y}=\frac{1}{1+\ln xr}$
$=\frac{1}{1+\ln x+\ln r}=\frac{1}{A+D}$
and $\frac{1}{1+\ln z}=\frac{1}{1+\ln x+2\ln r}$
$=\frac{1}{A+2D}$
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, $\frac{1}{1+\ln x},\frac{1}{1+\ln y},\frac{1}{1+\ln z}$, are in HP.