Question

If $\omega$ is a non-real cube root of unity, then $1+\omega +{\omega }^2+....+{\omega }^{101}$ is equal to

• A. $1$
• B. $\omega$
• C. ${\omega }^2$
• D. $0$

Answer 1

Answer: (D)

Given series $1+\omega +{\omega }^2+.....+{\omega }^{101}$ is a GM series
$\therefore$ $S=\frac{1({\omega }^{102}-1)}{\omega -1}$
$=\frac{1({({\omega }^2)}^{34}-1)}{\omega -1}=\frac{{(1)}^{34}-1}{\omega -1}(\because {\omega }^3=1)$
$=0$