Question 3
A3 (GRE) Let $a\gt 2$ be given, and define recursively
$$a_0=1,\qquad a_1=a,\qquad a_{n+1}=\left(\frac{a_n^2}{a_{n-1}^2}-2\right)a_n.$$
Show that for all $k\in\mathbb N$, we have
$$\frac1{a_0}+\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_k}\lt \frac12\left(2+a-\sqrt{a^2-4}\right).$$
$$a_0=1,\qquad a_1=a,\qquad a_{n+1}=\left(\frac{a_n^2}{a_{n-1}^2}-2\right)a_n.$$
Show that for all $k\in\mathbb N$, we have
$$\frac1{a_0}+\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_k}\lt \frac12\left(2+a-\sqrt{a^2-4}\right).$$