If $a_{1}, a_{2}, a_{3}$ are in A.P., $a_{2}, a_{3}, a_{4}$ are in G.P., and $a_{3}, a_{4}, a_{5}$ are in H.P., then $ln a_{1}, ln a_{3}, ln a_{5}$ are in

G.P.

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A.P.

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H.P.

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None of these

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Solution

The correct option is B A.P.
Given:
$a_{1}, a_{2}, a_{3}$ are in A.P.
$\Rightarrow 2 a_{2} = a_{1} + a_{3} \dots \dots (1)$
$a_{2}, a_{3}, a_{4}$ are in G.P.
$\Rightarrow a_{3}^{2} = a_{2} a_{4} \dots \dots (2)$
$a_{3}, a_{4}, a_{5}$ are in H.P.
$\Rightarrow a_{4} = \frac{2 a_{3} a_{5}}{a_{3} + a_{5}} \dots \dots (3)$
Using equation $(2)$ and $(3)$ ,
$\frac{a_{3}^{2}}{a_{2}} = \frac{2 a_{3} a_{5}}{a_{3} + a_{5}} \Rightarrow a_{3} (a_{3} + a_{5}) = 2 a_{2} a_{5}$
Using equation $(1)$ ,
$\Rightarrow a_{3}^{2} + a_{3} a_{5} = (a_{1} + a_{3}) a_{5} \Rightarrow a_{3}^{2} = a_{1} a_{5}$
Taking $ln$ on both sides,
$\Rightarrow 2 ln a_{3} = ln (a_{1} a_{5}) \Rightarrow 2 ln a_{3} = ln a_{1} + ln a_{5}$

Hence, $ln a_{1}, ln a_{3}, ln a_{5}$ are in A.P.

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