Question

Prove that $3^{\frac{1}{2}}\times 3^{\frac{1}{4}}\times 3^{\frac{1}{8}}.....=3$

Answer 1

Given,

$3^{\frac{1}{2}}\times 3^{\frac{1}{4}}\times 3^{\frac{1}{8}}\times 3^{\frac{1}{16}}\times ...............$

$=3^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.........}$

$=3^{{\sum }_{n=1}^{\infty }{\left(\frac{1}{2}\right)}^n}$

$=3^{\frac{\frac{1}{2}}{1-\frac{1}{2}}}$

$=3^1=3$

Hence proved.