The value of ω³ⁿ+ ω³ⁿ⁺¹ + ω³ⁿ⁺², where ω is cube roots of unity, is

Question

Question

Answer 1

Concept:

Cube Roots of unity are 1, ω and ω²

Here, ω = \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) and ω² = \(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\)

Property of cube roots of unity

Calculation:

We have to find the value of ω3n + ω3n+1 + ω³ⁿ⁺²

⇒ ω3n + ω3n+1 + ω³ⁿ⁺²

= ω3n (1 + ω + ω²) (∵ 1 + ω + ω2 = 0)

= 1 × 0 = 0