Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

$2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\dots \infty$
$=2^{\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\dots \infty }$
$=2^{\frac{1}{2^2}\left(1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\dots \right)}$
$=2^{\frac{1}{2^2}\left(\frac{1}{1-\frac{1}{2}}-\frac{1\cdot \frac{1}{2}}{{\left(1-\frac{1}{2}\right)}^2}\right)}$
$=2^{\frac{1}{2^2}|2+2|}=2^1=2$