First slide

Geometric progression

Question

Value of $y = (0.36)^{\log_{0.25} (\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \dots upto \infty})$ is

Moderate

A

$0.9$
B

$0.8$
C

$0.6$
D

$0.25$

Solution

We have
$S = \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \dots = \frac{1 / 3}{1 - 1 / 3} = \frac{1}{2}$
$\begin{array}{l} ∴ y = (0.36)^{\log_{a 25} (1 / 2)} \\ \Rightarrow \log y = \frac{\log (1 / 2)}{\log (0.25)} \log (0.36) \\ = \frac{- \log 2}{- \log 4} \log (0.36) = \frac{1}{2} \log (0.36) = \log 0.6 \\ \Rightarrow y = 0.6 \end{array}$

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