Let z = a+ib ( b 0 ) be a complex number such that the points representing z, z^2 , and z^3 form an equilateral triangle on the Argand plane. Then the value of _ k=1 ¹⁰ | z^k + 1 z^k |^2 is equal to _____ .

Step-by-step solution

Let the points be z₁ = z , z₂ = z^2 , and z₃ = z^3 . For an equilateral triangle, the vertices satisfy the condition: z₁^2 + z₂^2 + z₃^2 = z₁ z₂ + z₂ z₃ + z₃ z₁ Substituting the given points: z^2 + z^4 + z^6 = z^3 + z^5 + z^4 Rearranging the terms: z^6 - z^5 - z^3 + z^2 = 0 Since b 0 , z is non-zero. Dividing by z^2 gives: z^4 - z^3 - z + 1 = 0 Factoring the equation: z^3(z - 1) - 1(z - 1) = 0 (z^3 - 1)(z - 1) = 0 (z - 1)^2(z^2 + z + 1) = 0 Since b 0 , z cannot be purely real, so z 1 . Thus, z must satisfy: z^2 + z