Question

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

(a) n(n+1)a/2

(b) nb/(1-n)

(c) na/(ω-1)

(d) n(n+1)a/(ω-1)

I have been asked this question in an interview for job.

I'd like to ask this question from Roots of Complex Numbers in division Complex Numbers of Engineering Mathematics

Answer 1

The correct option is (c) na/(ω-1)

To explain I would say: (a+b)+(2a+b)ω+(3a+b)ω^2+…+(na+b)ω^n-1=b(1+ω+ω^2+….+ω^n-1)+a(1+2ω+3ω^2+…+nω^n-1)

Let S=1+2ω+3ω^2+…+nω^n-1⇒Sω= ω+2ω^2+…+(n-1)ω^n-1+nω^n⇒S(1-ω)=1+ω+ ω^2+…+ω^n-1–nω^n=-n

⇒S=n/(ω-1)⇒E=na/(ω-1).