If is a complex cube root of unity, then the value of the expression 2 (1+ 1 ) (1+ 1 ^2 ) + 3 (2+ 1 ) (2+ 1 ^2 ) + + (n+1) (n+ 1 ) (n+ 1 ^2 ) is...

Step-by-step solution

The general term of the given series is T_k = (k+1) (k+ 1 ) (k+ 1 ^2 ) . Since ^3 = 1 , we have 1 = ^2 and 1 ^2 = . T_k = (k+1)(k+ ^2)(k+ ) T_k = (k+1)(k^2 + k( + ^2) + ^3) Using 1+ + ^2 = 0 + ^2 = -1 and ^3 = 1 , we get: T_k = (k+1)(k^2 - k + 1) T_k = k^3 + 1 The sum of the series up to n terms is: S_n = _ k=1 ^n T_k = _ k=1 ^n (k^3 + 1) S_n = _ k=1 ^n k^3 + _ k=1 ^n 1 S_n = [ n(n+1) 2 ]^2 + n