Question

If $\frac{1}{4}\times \frac{2}{6}\times \frac{3}{8}\times \frac{4}{10}\times \frac{5}{12}\times .....\times \frac{31}{64}=\frac{1}{2^x}$, then what is the value of $x$?

• A. $6$
• B. $36$
• C. $42$
• D. $48$

Answer 1

The correct option is B $36$

$\frac{1}{2^x}=\frac{1}{4_2}\times \frac{2^1}{6_2}\times \frac{3^1}{{10}_2}\times \frac{5^1}{{12}_2}\times ......\times \frac{{30}^1}{{62}_2}\times \frac{{31}^1}{64}=\frac{1}{(2\times 2\times 2\text{times}...30\text{times})\times 64}$
$\Rightarrow \frac{1}{2^x}=\frac{1}{2^{30}\times 64}=\frac{1}{2^{30}\times 2^6}=\frac{1}{2^{36}}\Rightarrow x=36$