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  4. If ω is a non-real cube root of unity, then 1+ω +ω 2+....+ω 101 is equal to

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

J & K CETJ & K CET 2008Complex Numbers and Quadratic Equations

Solution:

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 $