Question

The value of $(0.2{)}^{{\log }_{\sqrt{5}}(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+....10n)}$ is

• A. $4$
• B. $\frac{1}{4}$
• C. $2$
• D. $\frac{1}{2}$

Answer 1

Answer: (A)

$4$

Let's calculate the sum of infinite series $\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+.......$

Let's consider it to be $S$

The series is a GP with first term $\left(a\right)=\frac{1}{4}$ and common ratio $\left(r\right)=\frac{1}{2}$

$\therefore S=\frac{a}{1-r}$

$\Rightarrow S=\frac{1}{2}$

To find

${\left(\frac{1}{5}\right)}^{{\log }_{\sqrt{5}}\frac{1}{2}}$

By Properties of log, we can write it as

${\left(\frac{1}{2}\right)}^{2{\log }_5\frac{1}{5}}$

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

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

$=2^2=4$

Hence, option A is correct.