Let , be the roots of the equation x^2 - (k^2 - 2k - 1)x + 5 = 0 , where k R . If a_n = ^n + ^n for n 1 , and it is given that a₁₀ + 5a₈ = 2a₉ with a₉ 0 , then the product of all possible values of k is

Step-by-step solution

By Newton's sums, for the quadratic equation x^2 - (k^2 - 2k - 1)x + 5 = 0 , we have the recurrence relation: a_ n+2 - (k^2 - 2k - 1)a_ n+1 + 5a_n = 0 Substituting n = 8 , we get: a₁₀ - (k^2 - 2k - 1)a₉ + 5a₈ = 0 a₁₀ + 5a₈ = (k^2 - 2k - 1)a₉ We are given that a₁₀ + 5a₈ = 2a₉ . Therefore, we can equate the two expressions: (k^2 - 2k - 1)a₉ = 2a₉ Since a₉ 0 , we can divide by a₉ to obtain: k^2 - 2k - 1 = 2 k^2 - 2k - 3 = 0 The roots of this quadratic equation in k are k = 3 and k = -1 . The product of all possible va