ISI2016-DCG-1

Question

The sequence $\dfrac{1}{\log_{3} 2},\dfrac{1}{\log_{6} 2},\dfrac{1}{\log_{12} 2},\dfrac{1}{\log_{24} 2}\cdots$ is in

• A. Arithmetic progression (AP)
• B. Geometric progression (GP)
• C. Harmonic progression (HP)
• D. None of these

Answer 1

Answer: $\mathbf{A}$

Explanation:

The series can be written as:

$\frac{1}{\log_32} = \log_23$

$\frac{1}{\log_62} = \log_26 = \log_2(2\times3) = \log_22+\log_23 = 1 + \log_23$

similarly,

$\frac{1}{\log_212} = \log_2(4\times3)= 2 + \log_23 $

$\frac{1}{\log_224} = \log_2(8\times3) = 3 + \log_23$

$\vdots$

$\vdots$

so on $\cdots$      $ \cdots $

So, we can clearly se that the terms are in $\mathbf{AP}$ with the common difference = $1$

$\therefore \mathbf{A}$ is the correct option.