If `n geq3` and `a_1,a_2,a_3,.....,a_(n-1)` are `n^(th)` roots of unity, then the sum of `sum_(1leqiltjleq(n-1)) a_i a_j`

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : 1+a_1+a_2+…+a_(n-1) =0.

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =n.

If a_1,a_2,a_3,.....,a_(n+1) be (n+1) different prime numbers, then the number of different factors (other than1) of a_1^m.a_2.a_3...a_(n+1) , is

If a_1,a_2,a_3,.....,a_(n+1) be (n+1) different prime numbers, then the number of different factors (other than1) of a_1^m.a_2.a_3...a_(n+1) , is

If a_1,a_2 ...a_n are nth roots of unity then 1/(1-a_1) +1/(1-a_2)+1/(1-a_3)..+1/(1-a_n) is equal to

If a_1,a_2,a_3,a_4 are the coefficients of 2nd, 3rd, 4th and 5th terms of (1+x)^n respectively then (a_1)/(a_1+a_2) , (a_2)/(a_2+a_3),(a_3)/(a_3+a_4) are in

Let 1/(a_1+omega) + 1/(a_2+omega)+1/(a_3+omega)+ … . + 1/(a_n+omega)=i where a_1,a_2,a_3 …. a_n in R and omega is imaginary cube root of unity , then evaluate sum_(r=1)^(n)(2a_r-1)/(a_r^2-a_r+1) .

Let 1/(a_1+omega) + 1/(a_2+omega)+1/(a_3+omega)+ … . + 1/(a_n+omega)=i where a_1,a_2,a_3 …. a_n in R and omega is imaginary cube root of unity , then evaluate sum_(r=1)^(n)(2a_r-1)/(a_r^2-a_r+1) .