question_answer

If \[\omega \] is an imaginary cube root of unity, then value of the expression \[2\,\,(1+\omega )\,\,(1+{{\omega }^{2}})+3\,\,(2+\omega )\] \[(2+{{\omega }^{2}})+.....+(n+1)\,\,(n+\omega )\,\,(n+{{\omega }^{2}})\] is

A)  \[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n+1)}^{2}}+n\]

B)  \[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n+1)}^{2}}-n\]

C)  \[\frac{1}{4}\,\,n\,\,{{(n+1)}^{2}}-n\,\]           

D)  \[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n-1)}^{2}}-n\]

E)  None of these