Question

If w is an imaginary cube root of unity, then find value of expression
2(1 + w) (1 + w^2) +3 (2 + w) (2 + w^2) + …+ (n + 1) (n + w) (n + w^2) in terms of n.

Answer 1

(k+w)(k+w^2) = w^3+kw^2+k^2w+k^3 = = (w^4-k^4)/(w-k)

See where this takes you:

https://www.wolframalpha.com/input/?i=sum(k%3D1..n)+(w%5E4-k%5E4)%2F(w-k)