If `omega` is a complex cube root of unity, then the value of the expression `1(2-omega)(2-omega^2)+2(3-omega)(3-omega^2) +...+(n-1) (n-omega)(n-omega^2) (n>=2) ` is equal to (A) `(n^2(n+1)^2)/4 - n` (B) `(n^2(n+1)^2)/4 +n` (C) `(n^2(n+1))/4 -n` (D) `(n(n+1)^2)/4 -n`

Explanation

We can write the given expression as ,

`S = sum_(m=1)^n(m-1)(m-omega)(m-omega^2)`

So , general term in this expression can be given as,

`T_m = (m-1)(m-omega)(m-omega^2)`

`= (m-1)(m^2-(omega+omega^2)m+omega^3)`

As, `1+omega+omega^2 = 0=>omega+omega^2 = -1`

`:. T_m = (m-1)(m^2+m+1)` (As `omega^3 = 1`)

We know, `(a-1)(a^2+a+1) = a^3-1`
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