Value of ω^n + ω^2n, where ω = (-1+i√3)/2 and n = 3k+1, is (a) 0 (b) -1 (c) 1 (d) None of these.

Value of ωⁿ + ω²n, where ω = (-1+i√3)/2 and n = 3k+1, is (a) 0 (b) -1 (c) 1 (d) None of these.

Question of 28

Q. Value of ω^n + ω^2n, where ω = (-1+i\sqrt3)/2 and n = 3k+1, is

(a) 0

(b) -1

(d) None of these.

West Bengal WbchseWest Bengal HS First Year (WBCHSE Class XI) Annual Examination 2018MCQ· 1mImportance★★★★★est

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$\omega$ is a primitive cube root of unity, so $\omega^n+\omega^{2n}$ reduces to $\omega+\omega^2=-1$ whenever $n=3k+1$ .

Here $\omega=\dfrac{-1+i\sqrt3}{2}$ is a complex cube root of unity, satisfying $\omega^3=1$ and $1+\omega+\omega^2=0$ .

Since $n=3k+1$ :

$\omega^n=\omega^{3k+1}=(\omega^3)^k\cdot\omega=1^k\cdot\omega=\omega$ …