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If nge3and1,alpha(1),alpha(2),alpha(3),....

If `nge3and1,alpha_(1),alpha_(2),alpha_(3),...,alpha_(n-1)` are
the n,nth roots of unity, then find value of`underset("1"le"i" lt "j" le "n" - "1" )(sum""sum) alpha _ "i" alpha _ "j" `

Text Solution

Verified by Experts

`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`
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