Q93 / 200 · Manipal (MET) 2011 · Mathematics

The value of $\log 2+2\left(\frac{1}{5}+\frac{1}{3} \cdot \frac{1}{5^3}+\frac{1}{5} \cdot \frac{1}{5^5}+\ldots.\right)$, is

  • A
    $\log 2+\log 3$
  • B
    $\log 2+2$
  • C
    $\frac{1}{2} \log 2$
  • Correct answer
    D
    $\log 3$

Solution

$\begin{aligned} \log 2+2 & \left(\frac{1}{5}+\frac{1}{3} \cdot \frac{1}{5^3}+\frac{1}{5} \cdot \frac{1}{5^5}+\ldots \infty\right) \\ & =\log 2+\log \left(\frac{1+\frac{1}{5}}{1-\frac{1}{5}}\right) \\ & =\log 2+\log \left(\frac{3}{2}\right)=\log 3\end{aligned}$

Hence, the correct answer is option: 'D'.

Relevant Questions