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If $ω$ is an imaginary cube root of unity, then the value of the expression $1 (2 - ω) (2 - ω^{2}) + 2 (3 - ω) (3 - ω^{2}) + \dots + (n - 1) (n - ω) (n - ω^{2})$ is

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$\frac{1}{4} n (n + 1)^{2} - n$

$\frac{1}{4} n^{2} (n + 1)^{2} - n$

$\frac{1}{4} n^{2} (n + 1)^{2} + n$

$\frac{1}{4} n^{2} (n + 1) - n$

answer is A.

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Detailed Solution

$\sum_{2}^{n} (n - 1) (n - ω) (n - ω^{2})$

${= \sum}_{2}^{n} (n^{3} - 1) = \frac{n^{2} (n + 1)^{2}}{4} - 1^{3} - [n - 1]$ = $\frac{1}{4} n^{2} (n + 1)^{2} - n$

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