Question

The value of the expression
$1(2-\omega)(2-\omega^2)+(3-\omega)(3-\omega^2)+......$
$ +(n-1).(n-\omega)(n-\omega^2),$
where, $\omega$ is an imaginary cube root of unity, is....

Answer 1

Here, $T_r= (r - 1) (r -\omega) (r - \omega)^2] $
$=(r^3-1)$
$\therefore S_n=\displaystyle \sum_{(r=1)}^n (r^3-1)=\bigg[\frac{n(n+1)}{2}\bigg]^2-n$