Question

The value of $2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}}\cdot 8^{\frac{1}{16}}\cdot {16}^{\frac{1}{32}}\dots \dots$ is

• A. $2$
• B. $\frac{3}{2}$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $2$

$2^{\frac{1}{4}}\cdot 4^{\frac{1}{8}}\cdot 8^{\frac{1}{16}}\cdot {16}^{\frac{1}{32}}\cdot \cdot \cdots$
$=2^{\frac{1}{4}}\cdot {\left(2^2\right)}^{\frac{1}{8}}\cdot {\left(2^3\right)}^{\frac{1}{16}}\cdot {\left(2^4\right)}^{\frac{1}{32}}\cdot \cdot \cdots$

$=2^{(\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\cdots )}=2^S$

$S=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+\cdots \to \left(1\right)$
$\frac{S}{2}=\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+\frac{4}{64}+\cdots \to \left(2\right)$
Subtracting equation $\left(2\right)$ from equation $\left(1\right),$ we get
$\frac{S}{2}=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\cdots$
$\Rightarrow \frac{S}{2}={\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^4+\cdots$
$\Rightarrow \frac{S}{2}=\frac{1/4}{1-1/2}=\frac{1}{2}$
$\Rightarrow S=1$

$\therefore$ Required value $=2^S=2^1=2$