Let n≥2 be an integer and z1,z2,z3 three complex numbers such that ∣zi∣=1,argzi∈[0,n2π)∀i∈{1,2,3} and z1n+z2n+z3n=0. Find ∣z1+z2+z3∣.[hide=My solution] I've used the fact that z1n,z2n,z3n are the affixes of an equilateral triangle, and finally I've got ∣z1+z2+z3∣=3+4cos3n2π+2cos3n4π.