Question

If '$\omega$' is a complex cube root of unity, then ${\omega }^{(\frac{1}{3}+\frac{2}{9}\frac{4}{27}+....\infty )}+{\omega }^{(\frac{1}{2}+\frac{3}{8}\frac{9}{32}+....\infty )}$

• A. $1$
• B. $-1$
• C. $\omega$
• D. $i$

Answer 1

The correct option is B $-1$

$(\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+....\infty )$(infinte G.P.)

$S_{\infty }=\frac{a}{1-r}=1$

$(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+....\infty )$ (infinte G.P.)

$S_{\infty }=\frac{a}{1-r}=2$

${\omega }^1+{\omega }^2=-1$