MathNet 2023-part-05 — 191 problems (EN–VI) Question 137

Find all real $\alpha \gt 0$ with the following property: there exists a sequence $(x_1, x_2, x_3, ...)$ of positive real numbers such that $x_{n+2} = \sqrt{\alpha x_{n+1} - x_n}$ for all $n \ge 1$.