If z is a complex number satisfying z + 1 z = 1 , then the value of | _ n=1 ²¹ ( z^n + 1 z^n )^3 | is equal to _____.

Step-by-step solution

Given z + 1 z = 1 z^2 - z + 1 = 0 . The roots of this equation are z = - and z = - ^2 , where is a non-real cube root of unity. Thus, z^3 = -1 . Let A_n = z^n + 1 z^n . Since z^3 = -1 , we have z^ n+6 = z^n , so A_n is periodic with period 6. Calculating the first six values of A_n : A₁ = z + 1 z = 1 A₂ = z^2 + 1 z^2 = (z + 1 z )^2 - 2 = 1 - 2 = -1 A₃ = z^3 + 1 z^3 = -1 - 1 = -2 A₄ = z^4 + 1 z^4 = -z - 1 z = -1 A₅ = z^5 + 1 z^5 = -z^2 - 1 z^2 = 1 A₆ = z^6 + 1 z^6 = 1 + 1 = 2 The terms in the summation are A_n^3 . F