Question

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

(A) 1

(B) $\sqrt{2}$

(C) $3 / 2$

(D) $5 / 2$

Answer 1

$2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \cdot 16^{1 / 32} \cdot \ldots \infty$

$=2^{\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots}$

$=2^{\frac{1}{4}\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots\right)} $

$=2^{\frac{1}{4}\left(\frac{1}{1-\frac{1}{2}}\right)}=\sqrt{2}$

So, The correct option will be (B).