Question

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

• A. $\frac{3}{2}$
• B. $\frac{5}{2}$
• C. $2$
• D. $1$

Answer 1

Answer: (C)

Let $S=2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot {16}^{1/32}$
$=2^{1/4}\cdot 2^{2/8}\cdot 2^{3/16}\cdot 2^{4/32}$
$=2^{\left(\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+...\right)}$
Let $S_1=1+2\cdot \frac{1}{2}+3\cdot \frac{1}{2^2}+4\cdot \frac{1}{2^3}+...$ .. (i)
$\therefore$ $\frac{1}{2}{S}_1=\frac{1}{2}+2\cdot \frac{1}{2^2}+3\cdot \frac{1}{2^3}+...$ .. (ii)
On subtracting Eq. (i) from Eq. (ii), we get
$\frac{1}{2}{S}_1=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...$
$=\frac{1}{1-\frac{1}{2}}$
$\Rightarrow$ $\frac{1}{2}{S}_1=2$
$\Rightarrow$ $S_1=4$
$\therefore$ $S=2^{\frac{1}{4}(4)}=2$