Question

If $a_1,a_2,a_3...$ are positive numbers in $GP$ then $\log a_n,\log a_{n+1},\log a_{n+2}$ are in

• A. $AP$
• B. $GP$
• C. $HP$
• D. none of these

Answer 1

The correct option is A $AP$

Since, $a_1,a_2,a_3,....$ are in G.P.
Let $a$ and $r$ be the first term and common ratio respectively
$\Rightarrow \log a_n=\log a+n\log r$ ...(1)
$\log a_{n+1}=\log a+(n+1)\log r$ ...(2)
and $\log a_{n+2}=\log a+(n+2)\log r$ ...(3)
From (1), (2) and (3)
$2\log a_{n+1}=\log a_n+\log a_{n+2}$
Therefore, $\log a_n,\log a_{n+1},\log a_{n+2}$ are in A.P.
Ans: A