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 the value of x is?

Solution(By Examveda Team)

$$\frac{1}{4} \times $$ $$\frac{2}{6} \times $$ $$\frac{3}{8} \times $$ $$\frac{4}{{10}} \times $$ $$\frac{5}{{12}} \times $$ . . . . . $$ \times \frac{{31}}{{64}}$$ $$ = \frac{1}{{{2^x}}}$$
Denominator of first term is cut by the numerator of next terms, second term to the next one and so on.
$$\eqalign{ & \Rightarrow {\left( {\frac{1}{2}} \right)^{30}} \times {\left( {\frac{1}{2}} \right)^6} = \frac{1}{{{2^x}}} \cr & \Rightarrow {\left( {\frac{1}{2}} \right)^{30 + 6}} = \frac{1}{{{2^x}}} \cr & \Rightarrow \frac{1}{{{2^{36}}}} = \frac{1}{{{2^x}}} \cr & \Rightarrow x = 36 \cr} $$