Question

If '$\omega$' is a complex cube root of unity,then ${\omega }^{\left(\begin{array}{cc}\frac{1}{3}& +\frac{2}{9}+\frac{4}{27}...\infty \end{array}\right)}+{\omega }^{\left(\begin{array}{cc}\frac{1}{2}& +\frac{3}{8}+\frac{9}{32}...\infty \end{array}\right)}=$

• 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.)
$S_{\omega }=\frac{a}{1-r}=1$
$\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+......+\infty$ (infinte G.P.)
$S_{\omega }=\frac{a}{1-r}=2$
$\therefore {\omega }^1+{\omega }^2=-1$