If ( ) represents a complex cube root of unity, then ( (1+ 1 ) (1+ 1 ^2 )+ (2+ 1 ) (2+ 1 ^2 ) ) (+ + (n+ 1 ) (n+ 1 ^2 )= )

Step-by-step solution

As ( ) is complex cube root of unity, then ( aligned & 1+ + ^2=0 and ^3=1 (i) & (r+ 1 ) (r+ 1 ^2 )=r^2+r ( 1 + 1 ^2 )+ 1 ^3 & =r^2+ ( ^2+ ^3 ) r+ 1 ^3 =r^2-r+1 [from Eq. (i)] & Now, _ r=1 ^n (r+ 1 ) (r+ 1 ^2 ) & = _ r=1 ^n (r^2-r+1 )= n(n+1)(2 n+1) 6 - n(n+1) 2 +n & = n(n+1) 6 [2 n+1-3]+n= n(n+1) 6 (2 n-2)+n & = n 3 (n^2-1 )+n= n 3 [n^2-1+3 ]= n (n^2+2 ) 3 aligned ) Hence, option (2) is correct.