Question

Assertion :Let $2^{1/14}{.4}^{1/8}{.8}^{1/16}{.16}^{1/32}...\infty =2^x$ then the value of $x$ is equal to 1. Reason: The sum of series $\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}=...\infty$ is equal to 1.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$2^{1/4}{4}^{1/8}{8}^{1/16}(16{)}^{1/32}...\infty =2^x$
$\Rightarrow 2^x=2^{\frac{1}{4}}{2}^{\frac{2}{8}}{2}^{\frac{3}{16}}{2}^{\frac{4}{32}}...\infty =2^{(\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+...\infty )}$
$\Rightarrow x=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+...\infty$ ...............(i)
$\Rightarrow \frac{x}{2}=\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+...\infty$ .......(ii)
Subtracting (i) and (ii), we get
$\frac{x}{2}=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+...\infty$
$\Rightarrow \frac{x}{2}=\frac{1}{4}\left(\frac{1}{1-\frac{1}{2}}\right)$
Therefore, $x=1$
Ans: A