Doubt from complex numbers

plz solve question 9 answer is d option.

Write the first three initial and final terms, it'll be easier to visualise, you'll see that the last and first,second last and second terms have a common factor and so on,adding all these clubbed terms will result in the answer as you'd be able to take the number 2009 common and the other factor would be the sum of the two thousand and ninth roots of unity except for 1

Thnx @Raunak_2020 here is a way in which I solved, Clarify if needed



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