If ( ) is a complex cube root of unity, then ( _ r=1 ^9 r(r+1- ) (r+1- ^2 )= )

Step-by-step solution

As, ( ) is complex cube root of unity, then ( aligned 1+ + ^2 & =0 and ^3 & =1 (i) aligned ) ( aligned & r(r+1- ) (r+1- ^2 ) & =r [(r+1)^2- ( + ^2 )(r+1)+ ^3 ] & =r [(r+1)^2+(r+1)+1 ] [from Eq. (i)] & =r (r^2+3 r+3 )=r^3+3 r^2+3 r & _ r=1 ^9 r(r+1- ) (r+1- ^2 )= _ r=1 ^9 (r^3+3 r^2+3 r ) & = ( 9(10) 2 )^2+3 9(10)(19) 6 +3 9 10 2 & .=(45)^2+(45 19)+(3 45)=4545+19+3 ] & =45 67=3015 aligned ) Hence, option (4) is correct.