Question

If 1,alpha1..... Are the roots of unity, then the value of is:

Answer 1

1,α1,α2,…,αn are the nth of unity. These are the roots of xn−1=0

Let y=11−α where α=α1,α2,α3,…,αn

1−α=1y⇒α=y−1y.

But α is a root of xn−1=0∴αn=1⇒(y−1)n=yn

⇒yn−nC1yn−1+nC2yn−2−…+(−1)n=yn⇒−nC1yn−1+nC2yn−2−…+(−1)n=0.

Sum of roots

11−α1+11−α2+…+11−αn−1=nC2nC1=n−12