2011/047) the value of the expression

2(1+w)(1+w^2) + 3(2w+1)(2w^1+1) + ... (n+1)(nw+1)(nw^2+1) when w is cube root of unit is

1) n^2 (n + 1)^2/4
2) n^2 (n + 1)^2/4 -n
3) n^2 (n + 1)^2/4 + n
4_ no of these

ans 3) because

kth term = (k+1)(kw+1)(kw^2+1) = (k+1) (k^2w^3+ k(w+w^2) + 1)
= (k+1)(k^2 -k +1) as w^3 = 1 and w + w^2- 1
= (k^3 + 1)

so sum = (sum of cubes upto n) + n
= (n(n+1)/2)^2 + n
= n^2(n+1)^2 /4 + n