MathNet 2015-part-05 — 191 problems (EN–VI) Question 139

Let $a_1, a_2, a_3, \dots$ be a sequence of non-zero real numbers which satisfies $a_n^2 = -a_{n+1}a_{n-1}$ for all positive integers $n$, $n \ge 2$. Prove that the sequence $a_2, a_4, a_6, \dots$ is geometric.