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  4. If ω is an imaginary cube root of unity, then the value of (2-ω)(2-ω2)+2(3-ω)(3-ω2)+.....+(n-1)(n-ω)(n-ω2)

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Q. If $\omega$ is an imaginary cube root of unity, then the value of $\left(2-\omega\right)\left(2-\omega^{2}\right)+2\left(3-\omega\right)\left(3-\omega^{2}\right)+.....+\left(n-1\right)\left(n-\omega\right)\left(n-\omega^{2}\right)$

WBJEEWBJEE 2016Complex Numbers and Quadratic Equations

Solution:

$\displaystyle \sum_{r=2}^n\left(r-1\right)\left(r-\omega^{2}\right)=\displaystyle \sum_{r=2}^n(r^3-1)=$$\left[\frac{n^{2}\left(n+1\right)^{2}}{4}-1\right]-\left(n-1\right)=\frac{n^{2}\left(n+1\right)^{2}}{4}-n$