Question

$2^{1 / 4} \cdot 2^{2 / 8} \cdot 2^{3 / 16} \cdot 2^{4 / 32} \ldots \infty$ is equal to -

• A. 1
• B. 2
• C. 3 / 2
• D. 5 / 2

Answer 1

Answer: (B)

The given product $=2^{\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+\ldots .}=2^{ s }$ (say)
Now $S =\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+..... \,\,\,\, \, \dots(1)$
$\Rightarrow \frac{1}{2} S =\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+\ldots\,\,\,\,\, \dots(2)$
$\therefore (1)-(2)$
$\Rightarrow \frac{1}{2} S =\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots$
$=\frac{1 / 4}{1-1 / 2}=\frac{1}{2}$
$\therefore S =1$
$\Rightarrow $ Product $=2^{1}=2$