Question

Let a${}_1$, a${}_2$, a${}_3$, a${}_4$ and a${}_5$ be such that a${}_1$, a${}_2$ and a${}_3$ are in A.P., a${}_2$, a${}_3$ and a${}_4$ are in G.P., and a${}_3$, a${}_4$ and a${}_5$ are in H.P. Then log${}_e$ a${}_1$, log${}_e$$a_3$ and log${}_e$ a${}_5$ are in

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

Answer 1

The correct option is B

A.P.

We have,

$a_1,a_2,a_3$ are in A.P. $\Rightarrow 2a_2=a_1+a_3$

$a_2,a_3,a_4$ are in G.P. $\Rightarrow a_3^2=a_2{a}_4$

$a_3,a_4,a_5$ are in H.P. $\Rightarrow a_4=\frac{2a_3{a}_5}{a_3+a_5}$

Putting $a_2=\frac{a_1+a_3}{2}$ and $a_4=\frac{2a_3{a}_5}{a_3+a_5}$ in (2)we get

$a_3^2=\frac{a_1+a_3}{2}\times \frac{2a_3{a}_5}{a_3+a_5}$

$\Rightarrow a_3^2=a_1{a}_5$

Hence, $a_1,a_3,$ and $a_5$ are in G.P. So $lo{g}_e$ $a_1$ $lo{g}_e$ $a_3$ and $lo{g}_e$ $a_5$ are in A.P.