If 1 + x 2 = 3 x , then ∑ n = 1 24 x n - 1 x n 2 is equal to

Step-by-step solution

Given, x 2 - 3 x + 1 = 0 ⇒       x = 3 ± 3 - 4 2 = 3 ± i 2 = cos ⁡ π 6 ± i sin ⁡ π 6   ⇒       x n = cos ⁡ n π 6 ± i sin ⁡ n π 6 And 1 x n = cos ⁡ n π 6 ± i sin ⁡ n π 6 ∴       x n - 1 x n = ± 2 i sin n π 6 ⇒       x n - 1 x n 2 = - 4 sin 2 ⁡ n π 6 i 2 = - 1 Hence, ∑ n = 1 24 x n - 1 x n 2 = - 4 sin 2 ⁡ &#