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Find ∑r=1(ar+b) ωr-1 if ω is a complex nth root of unity.

Find ∑r=1(ar+b) ωr-1 if ω is a complex nth root of unity. Correct Answer na/(ω-1)

(a+b)+(2a+b)ω+(3a+b)ω2+…+(na+b)ωn-1=b(1+ω+ω2+….+ωn-1)+a(1+2ω+3ω2+…+nωn-1) Let S=1+2ω+3ω2+…+nωn-1⇒Sω= ω+2ω2+…+(n-1)ωn-1+nωn⇒S(1-ω)=1+ω+ ω2+…+ωn-1–nωn=-n ⇒S=n/(ω-1)⇒E=na/(ω-1).

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