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IPMAT - Indore
IPMAT (I) 2020 QASA

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Question:

The value of $0 . 04^{\log_{\sqrt{5}} (\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + . . .)}$ is _________.

Explanation:

Given, $0 . 04^{\log_{\sqrt{5}} (\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + . . .)}$

= $0 . 04^{\log_{\sqrt{5}} (\frac{1 / 4}{1 - 1 / 2})}$

= $0 . 04^{\log_{\sqrt{5}} (\frac{1}{2})}$

= ${(5^{- 2})}^{\log_{\sqrt{5}} (\frac{1}{2})}$

= ${(5^{- 2})}^{\log_{5^{1 / 2}} (\frac{1}{2})}$

= ${(5^{- 2})}^{2 \log_{5} (\frac{1}{2})}$

= $5^{- 4 \log_{5} (\frac{1}{2})}$

= $5^{\log_{5} {(\frac{1}{2})}^{- 4}}$

= $5^{\log_{5} 16}$

= $16^{\log_{5} 5}$

= 16.

Hence, 16.

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