MathNet 2015-part-05 — 191 problems (EN–VI) Question 183

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of nonzero real numbers such that $a_{n}^{2} = -a_{n+1} a_{n-1}$ for all natural numbers $n$, $n \geq 2$. Prove that the sequence $a_{2}, a_{4}, a_{6}, \ldots$ is geometric.