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Q.
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 :
Sequences and Series
Solution:
$x, y, z$ are in GP
$
\Rightarrow y ^{2}= xz
$
Taking log on both sides
$
\begin{array}{l}
\Rightarrow 2 \ln y=\ln x+\ln z \\
\Rightarrow 2(1+\ln y)=(1+\ln x)+(1+\ln z)
\end{array}
$
i.e $1+\ln x , 1+\ln y , 1+\ln z$ are in $AP$
$\therefore \frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z}$ are in HP.