For a complex number Z , Z = 1 and arg Z = θ . If Z Z 2 Z 3 . . . . Z n = 1 , then the value of θ is

Step-by-step solution

Let, Z = e i θ ,   Z 2 = e i 2 θ , . . . . . . . . . . , Z n = e i n θ Given, Z Z 2 . . . . . . . Z n = 1 ⇒ e i θ e i 2 θ . . . . . . . e i n θ = 1 ⇒ e i θ 1 + 2 + . . . . + n = e i 2 m π   m ∈ Ι ⇒ θ n n + 1 2 = 2 m π ,   m ∈ Ι ⇒ θ = 4 m π n n + 1 ,   m ∈ Ι