The value of the expression `1.(2-omega).(2-omega^2)+2.(3-omega)(3-omega^2)+.+(n-1)(n-omega)(n-omega^2),` where omega is an imaginary cube root of unity, is………

Explanation

`S=1xx(2-omega)xx(2-omega^(2))+2xx(3-omega)xx(3-omega)xx(3-omega^(2))+…+(n-1)xx(n-omega)xx(n-omega^(2))`
Here, `T_(n)=(n-1)xx(n-omega)xx(n-omega^(2))=n^(3)-1`
`thereforeS=sum_(n=2)^(n)(n^(3)-1)`
`=sum_(n=1)^(n)(n^(3)-1)`
`=(n^(2)(n+1)^(2))/4-n`
`=(n^(2)(n^(2)+2n+1)-4n)/4`
`1/4n(n^(3)+2n^(2)+n-4)`
`=1/2n(n-1)(n^(2)+3n+4)`