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, $y=x r$ 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, $\frac{1}{1+\ln x}=\frac{1}{A}$
$\frac{1}{1+\ln y}=\frac{1}{1+\ln x r}$
$=\frac{1}{1+\ln x+\ln r}=\frac{1}{A+D}$
and $\frac{1}{1+\ln z}=\frac{1}{1+\ln x+2 \ln r}$
$=\frac{1}{A+2 D}$
Here we see that $A, A+D$ and $A+2 D$ are in AP.
$\therefore \frac{1}{A}, \frac{1}{A+D}$ and $\frac{1}{A+2 D}$ are in HP.
Therefore, $\frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z}$, are in HP.