If is a complex cube root of unity, then for any n>1 , _ r=1 ^ n-1 r(r+1- ) (r+1- ^2 )=

Step-by-step solution

We have, aligned & = _ r=1 ^ n-1 r(r+1-w) (r+1-w^2 ) & = _ r=1 ^ n-1 r [(r+1)^2-(r+1) w^2-w(r+1)+w^3 ] & = _ r=1 ^ n-1 r [(r+1)^2-(r+1) ( (w+w^2 )+w^3 ) ] & = _ r=1 ^ n-1 r [(r+1)^2-(r+1)((-1)+1) ] & = _ r=1 ^ n-1 r (r^2+3 r+3 ) [ w^3=1 and 1+w+w^2=0 ] & = _ r=1 ^ n-1 r^3+3 _ r=1 ^ n-1 r^2+3 _ r=1 ^ n-1 r & = ( (n-1) n 2 )^2+ 3(n-1)(n)(2 n-1) 6 + 3(n-1) n 2 & n^2(n-1)^2 4 + n(n-1)(2 n-1) 2 + 3 2 n(n-1) & = n(n-1) 4 [n(n-1)+2(2 n-1)+6] & = n(n-1) 4 [n^2-n+4 n-2+6 ] aligned = n(n-1) 4 [n^2+3 n+4 ]