Question

The product of the numbers $2^{1/4}\cdot 4^{1/8}\cdot 8^{1/16}\cdot 16^{1/32}$ ... is

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

Answer 1

Answer: (D)

Product $=2^{1/4}{2}^{2/8}{2}^{3/16}{2}^{4/32}{2}^{5/64}...$
$=2^{\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+\frac{5}{64}+....\infty }....\left(\ast \right)$
$P=2^R$, where $R=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+....\left(i\right)$
$\therefore \frac{R}{2}=\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+...\left(ii\right)$
$\therefore$ (i) -(ii) gives
$R-\frac{R}{2}=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+...=\frac{1/4}{1-1/2}$
or$\frac{R}{2}=\frac{1}{4}\times \frac{2}{1}$ or $R=1$
$\therefore$ Product $=2^R=2^1=2$