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4 Answers

6 votes
6 votes
As $\frac{1}{log_nk} = log_kn$ the sequence can be written as $log_23,log_26,log_212$ and so on. According to AP formula a,b,c are in AP iff b-a = c-b. Now in the given series $log_26 - log_23 = log_2{\frac{6}{3}} = 1$ and  $log_212 - log_26 = log_2{\frac{12}{6}} = 1$ and so on. So the given sequence is in AP. Option A is correct.
3 votes
3 votes

$\log_{a}{b} \times \log_{b}{a}=\log_{a}{a}=1;~[\because \log_{a}{b} \times \log_{b}{c} = \log_{a}{c}\mathrm{~and~}\log_{k}{k}=1]$

$\therefore \log_{a}{b}=\frac{1}{\log_{b}{a}}$

Again we know that $\log_{k}{(ab)}=\log_{k}{a}+\log_{k}{b}$ and $\log_{k}{(a^r)}=r\log_{k}{a}$.

So we can derive $\log_{a}{a^r}=r\log_{a}{a}=r\times 1=r$

Now the progression is


$\begin{align} &~~~~\frac{1}{\log_{3}{2}},\frac{1}{\log_{6}{2}},\frac{1}{\log_{12}{2}},\frac{1}{\log_{24}{2}},\cdots \\ &\equiv \log_{2}{3},~\log_{2}{6},~\log_{2}{12},~\log_{2}{24},\cdots \\ &\equiv \log_{2}{3},~\log_{2}{(2\times3)},~\log_{2}{(2^2\times3)},~\log_{2}{(2^3\times3)},\cdots \\ &\equiv \log_{2}{3},~\log_{2}{2}+\log_{2}{3},~\log_{2}{2^2}+\log_{2}{3},~\log_{2}{2^3}+\log_{2}{3},\cdots \\ &\equiv \log_{2}{3},~1+\log_{2}{3},~2+\log_{2}{3},~3+\log_{2}{3},\cdots \end{align}$

 

Clearly $\log_{2}{3},~1+\log_{2}{3},~2+\log_{2}{3},~3+\log_{2}{3},\cdots$ is an Arithmetic progression because the difference between two consecutive terms is $1$.

 

So the correct answer is A.

 

0 votes
0 votes
$log_{a} \ b = log_{b} \ b \ / log_{b} \ a = 1 \ / log_{b} \ a$

$1 \ / log_{3} \ 2 = log_{2} \ 3$

 

$1 \ / log_{6} \ 2 = log_{2} \ 6$

 

$1 \ / log_{12} \ 2 = log_{2} \ 12$

 

$1 \ / log_{24} \ 2 = log_{2} \ 24$

 

$log_{2} \ 3 , log_{2} \ 6 , log_{2} \ 12, log_{2} \ 24$

 

$log_{2} \ 6 - log_{2} \ 3 = log_{2} \ 2$                             $\because log \ b - log \ a = log (b/a)$

 

$log_{2} \ 12 - log_{2} \ 6 = log_{2} \ 2$

 

$log_{2} \ 24 - log_{2} \ 12 = log_{2} \ 2$

 

same $common \  difference$ , so it is in $A.P$

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