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

$\begin{array}{1 1}(A)\;\large\frac{1}{4}\normalsize n^2(n+1)^2-n&(B)\;\large\frac{1}{4}\normalsize n^2(n+1)^2+n\\(C)\;\large\frac{1}{4}\normalsize n^2(n-1)^2-n&(D)\;\large\frac{1}{4}\normalsize n^2(n-1)^2+n\end{array} $

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$r^{th}$term=$(r)(r+1-\omega)(r+1-\omega^2)$
$\Rightarrow (r+1-1)(r+1-\omega)(r+1-\omega^2)$
$\Rightarrow (r+1)^3-1$
$\Rightarrow \sum\limits_{r=0}^{n-1}(r+1)^3-1$
$\Rightarrow \sum\limits_{r=1}^n r^3-n$
$\Rightarrow \large\frac{1}{4}$$n^2(n+1)^2-n$
Hence (A) is the correct answer.
answered Apr 10, 2014 by sreemathi.v
 

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