Question

$2^{\frac{1}{4}}$. $4^{\frac{1}{8}}$. $8^{\frac{1}{16}}$. ${16}^{\frac{1}{32}}$................ is equal to

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

Answer 1

The correct option is B

2

$2^{\frac{1}{4}}$. $4^{\frac{1}{8}}$. $8^{\frac{1}{16}}$. ${16}^{\frac{1}{32}}$ .........∞

= $2^{\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+........}$ = $2^S$, where S is given by

S = $\frac{1}{4}$ + 2 $\frac{1}{8}$ + 3 $\frac{1}{16}$ +4 $\frac{1}{32}$ + ............∞ .........(i)

⇒ $\frac{1}{2}$ S = $\frac{1}{8}$ + $\frac{2}{16}$ + $\frac{3}{32}$ + $\frac{4}{64}$ + ................∞ ..........(ii)

Subtracting (ii) from (i), we get S = 1.

Hence required product = $2^1$ = 2.