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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:


A
A.P.
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B
G.P.
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C
H.P.
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D
none of these
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Solution

The correct option is A A.P.
$$x_{1}, y_{1}, z \rightarrow G \cdot Y$$
$$y^{2}=x \times z$$
taking log both sides
$$\ln y^{2}=\ln (x \times z)$$
$$2 \ln y=\ln x+\ln z$$
$$2(1+\ln y)=(1+\ln x)+(1+\ln z)$$
$$\therefore \quad 1+\ln x, \quad 1+\ln y, \quad 1+\ln z$$ are in $$ AP$$
Hence $$\dfrac{1}{1+\ln x}, \dfrac{1}{1+\ln y}, \dfrac{1}{1+\ln z}$$ are in H.P.

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