First slide

Geometric progression

Question

The value of ${(0 \cdot 2)}^{\log_{\sqrt{5}} (\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots)}$ is

Moderate

A

4
B

log 4
C

log 2
D

none of these

Solution

We have,
$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = \frac{1 / 4}{1 - 1 / 2} = \frac{1}{2}$
Hence,
$0 {.2}^{\log_{\sqrt{5}} (\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots)} = 0 {.2}^{\log_{\sqrt{5}} \frac{1}{2}} = {(\frac{1}{5})}^{\log_{\sqrt{5}} \frac{1}{2}} \begin{array}{l} = {(5^{- 1})}^{2 \log 5 \frac{1}{2}} \\ = (5)^{- 2 \log_{5} \frac{1}{2}} \\ = (5)^{\log_{5} 4} \\ = 4 \end{array}$

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