If `n ge 3` and `,alpha_(1),alpha_(2)......,alpha_(n-1)` are nth roots of unity , then find the sum `sum_(1leilejle n-1) alpha_(i)lpha_(j)`

Explanation

`sum _(1le i le j le n-1) alpha_(i) alpha_(j)` =sum of the proudct of (n-1) complex roots taken two at a time
Now `(alpha_(1) + alpha_(2) +........+alpha_(n-1))^(2)`
`= 2(sum_(1le ile j le n-1) alpha_(i)alpha_(j))+ (alpha_(1)^(2))+(alpha_(2))^(2) +.....+(alpha _(n-1) )^(2)`
`therefore (-1)^(2) =2(sum_(1le ile j le n-1) alpha_(i)alpha_(j) )-1`
`rArr sum_(1le ile jle n-1) alpha_(i)alpha_(j) = 1`