Let and be the roots of the equation x^2-x+1=0 . If _ k =1 ^ n ( ^ k + ^ k ) = 1024 , then the smallest positive integer n is

Step-by-step solution

The roots of x^2-x+1=0 are = - and = - ^2 , where is the complex cube root of unity. Let x_k = ^k + ^k = (- )^k + (- ^2)^k = (-1)^k ( ^k + ^ 2k ) . We evaluate the first few terms of the sequence x_k : k=1: x₁ = (-1)^1 ( + ^2) = (-1)(-1) = 1 k=2: x₂ = (-1)^2 ( ^2 + ^4) = (1)(-1) = -1 k=3: x₃ = (-1)^3 ( ^3 + ^6) = (-1)(2) = -2 k=4: x₄ = (-1)^4 ( ^4 + ^8) = (1)(-1) = -1 k=5: x₅ = (-1)^5 ( ^5 + ¹⁰) = (-1)(-1) = 1 k=6: x₆ = (-1)^6 ( ^6 + ¹²) = (1)(2) = 2 The sequence repeats with a period of 6. The product of the first