First slide

Geometric progression

Question

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

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

As $x, y, z$ are in G.P. $\Rightarrow y^{2} = x z$
$\begin{array}{l} \Rightarrow \ln (y^{2}) = \ln (x z) \Rightarrow 2 \ln (y) = \ln (x) + \ln (z) \\ \Rightarrow 2 (1 + \ln (y)) = (1 + \ln (x)) + (1 + \ln (z)) \\ \Rightarrow 1 + \ln (x), 1 + \ln y), 1 + \ln (z) are in A.P. \\ \Rightarrow \frac{1}{1 + \ln (x)}, \frac{1}{1 + \ln (y)}, \frac{1}{1 + \ln (z)} are in H.P. \end{array}$

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