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Question: \(\lim_{n \rightarrow \infty}(0.2)^{\log_{\sqrt{5}}(1/4 + 1/8 + 1/16 + ...nterms)}\)is equal to-...

lim⁡n→∞(0.2)log⁡5(1/4+1/8+1/16+...nterms)\lim_{n \rightarrow \infty}(0.2)^{\log_{\sqrt{5}}(1/4 + 1/8 + 1/16 + ...nterms)}is equal to-

A

2

B

4

C

8

D

0

Answer

4

Explanation

Solution

lim⁡n→α(0.2)log⁡5[14((1−12n)1−12)]\lim_{n \rightarrow \alpha}(0.2)^{\log_{\sqrt{5}}\left\lbrack \frac{1}{4}\left( \frac{\left( 1 - \frac{1}{2^{n}} \right)}{1 - \frac{1}{2}} \right) \right\rbrack} = (0.2)log⁡5[12(1−1α)](0.2)^{\log_{\sqrt{5}}\left\lbrack \frac{1}{2}\left( 1 - \frac{1}{\alpha} \right) \right\rbrack}

= 15log⁡5(12)\frac{1}{5}^{\log_{\sqrt{5}}\left( \frac{1}{2} \right)}= 5log⁡5(12)−15^{{\log_{\sqrt{5}}\left( \frac{1}{2} \right)}^{- 1}}

= 5log⁡5(2)5^{\log_{\sqrt{5}}(2)}Ž 511/2log⁡525^{\frac{1}{1/2}\log_{5}2}Ž 52log⁡525^{2\log_{5}2}= 5log⁡5(2)25^{\log_{5}(2)^{2}}= 4