If `alpha` is an `n^(th)` root of unity then prove that `1+2alpha+3alpha^2+4alpha^3+….+nalpha^(n-1)=-(n)/(1-alpha)`

If 1,alpha,alpha^2,alpha^3,…,alpha^(n-1) be the n^(th) roots of unity, then prove that 1^p+alpha^p+(alpha^2)^p+(alpha^3)^p+…+(alpha^(n-1))^p={{:(0,if" "pnekn),(n,if" "p=kn):}" where p,k"epsilonN

If alpha is a non-real root of x^(7) = 1 then alpha (1 + alpha)(1 + alpha^(2) + alpha^(4))=

If 'alpha' be the non-real n^(th) roots of unity, then 1+3alpha+5alpha^2+…...(2n-1)alpha^(n-1) is equal to

If 1,alpha,alpha^2,…...,alpha^(n-1) are the n^(th) roots of unity, then the value of (3-alpha)(3-alpha^2)…...(3-alpha^(n-1)) is

If alpha is a non-real root of x^(7) = 1 then alpha(1 + alpha) (1 + alpha^(2) + alpha^(4)) =

If alpha is a non real root of x^6=1 then (alpha^5+alpha^3+alpha+1)/(alpha^2+1)=

If alpha is a non real root of the equation x^(6)-1=0 then (alpha^(2)+alpha^(3)+alpha^(4)+alpha^(5))/(alpha+1)

IF alpha is a root of the equation 4x ^2 +2x-1=0 then 4 alpha ^3 - 3 alpha is

If alpha, beta are the roots of x^(2) + x + 3=0 then 5alpha+alpha^(4)+ alpha^(3)+ 3alpha^(2)+ 5beta+3=