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  4. The value of (0.2) log√5((1/4) + (1/8) + (1/6) + .... to ∞)

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Q. The value of $\left(0.2\right)^{\log_{\sqrt{5}}\left(\frac{1}{4} + \frac{1}{8} + \frac{1}{6} + .... to \, \infty\right)}$

COMEDKCOMEDK 2009Probability - Part 2

Solution:

$0.2^{\log_{\sqrt{5}}\left(\frac{1}{4} + \frac{1}{8} + \frac{1}{6} + .... to \infty\right)} $
$= \left(\frac{1}{5}\right)^{\log _{\sqrt{5}} \left(\frac{1}{2}\right)} = \left(\frac{1}{5}\right)^{2 \log _{\sqrt{5}}}\left(\frac{1}{2}\right)$
$ = \left(5\right)^{-2\log _{5}} \left(\frac{1}{2}\right) = \left(5\right)^{\log _{5}4} = 4 $