Question

Let $z =\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1}$. Then the value of $21+\left(z+\frac{1}{z}\right)^{3}+\left(z^{2}+\frac{1}{z^{2}}\right)^{3}+\left(z^{3}+\frac{1}{z^{3}}\right)^{3}+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^{3}$ is _____

Answer 1

$Z =\frac{1-\sqrt{3} i }{2}= e ^{- i \frac{\pi}{3}}$
$z ^{ r }+\frac{1}{ z ^{ r }}=2 \cos \left(-\frac{\pi}{3}\right) r =2 \cos \frac{ r \pi}{3}$
$\Rightarrow 21+\displaystyle\sum_{ r =1}^{21}\left( z ^{ r }+\frac{1}{ z ^{ r }}\right)^{3}=8\left(\cos ^{3} \frac{ r \pi}{3}\right)=2\left(\cos r \pi+3 \cos \frac{ r \pi}{3}\right)$
$\Rightarrow 21+\left( z +\frac{1}{2}\right)^{3}+\left( z ^{2}+\frac{1}{ z ^{2}}\right)^{3}+\ldots \ldots\left( z ^{21}+\frac{1}{ z ^{21}}\right)^{3}$
$=21+\displaystyle\sum_{ r =1}^{21}\left( z ^{ r }+\frac{1}{ z ^{ r }}\right)^{3}$
$=21+\displaystyle\sum_{ r =1}^{21}\left(2 \cos r \pi+6 \cos \frac{ r \pi}{3}\right)$
$=21-2-6$
$=13$