Question

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 $
• D. $ \log 3 $

Answer 1

Answer: (D)

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