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Q.
If $ z+{{z}^{-1}}=1 $ , then $ {{z}^{100}}+{{z}^{-100}} $ is equal to
Jharkhand CECEJharkhand CECE 2011
Solution:
$z+z^{-1}=1 \Rightarrow z^{2}-z+1=0$
$\Rightarrow z=-\omega$ or $-\omega^{2}$ For $z=-\omega$, we have
$z^{100}+z^{-100}=(-\omega)^{100},(-\omega)^{-100}$
$=\omega+\frac{1}{\omega}=\omega+\omega^{2}=-1$
For $z=-\omega^{2}$
$z^{100}+z^{-100}=\left(-\omega^{2}\right)^{100}+\left(-\omega^{2}\right)^{-100}$
$=\omega^{200}+\frac{1}{\omega^{200}}=\omega^{2}+\frac{1}{\omega^{2}}=\omega^{2}+\omega=-1$