question_answer

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

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