Question

The value of $0.2\sqrt[log]{log5}(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...)$ is

• A. 4
• B. log 4
• C. log 2
• D. log 2

Answer 1

The correct option is A

4

We have,

$\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+.....=\frac{1/4}{1-1/2}=\frac{1}{2}$

Hence,

$0.2\sqrt[log]{log5}(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...)=0.2\sqrt[log]{log5}\frac{1}{2}$

$=\left(\frac{1}{5}\right)\sqrt[log]{log5}\frac{1}{2}$

$={\left(5^{-1}\right)}^{2log5\frac{1}{2}}$

$={\left(5\right)}^{-2log5\frac{1}{2}}$

$={\left(5\right)}^{log{5}^4}$

$=4$