WBJEE Question

it is A.....

ketan pls explain it...

Its an A.G.P. series
S=1+2ω+3ω²+......+3nω^3n-1 ......(1)
Sω=ω+2ω²+........+(3n-1)ω³ⁿ+3nω³ⁿ ......(2)
(1)-(2) is
S(1-ω)=1+ω+ω²+.......+ω^3n-1 -3nω³ⁿ
after that wat ???
pls explain.......

since 1,w,w² are roots of x³=1 ,
sum of roots =0
or, 1+w+w²=0
similarly, w³+w⁴+w⁵=0 ..... and so on

thus S(1-w)= 0 +0+....0 - 3n (as w³=1)
or S= 3nw-1

another way to do it,
consider the function f(x) = x + x²+x³....+x³ⁿ
we are required to find f'(w) .

thnx rishabh and Aditya..........