Question 7
Let $a \gt 2$ be given, and define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ by
$$
a_{0} = 1, \quad a_{1} = a, \quad a_{n+1} = \left(\frac{a_{n}^{2}}{a_{n-1}^{2}} - 2\right) \cdot a_{n} .
$$
Show that for all integers $k \geq 0$, we have
$$
\sum_{i=0}^{k} \frac{1}{a_{i}} \lt \frac{a+2-\sqrt{a^{2}-4}}{2} .
$$
$$
a_{0} = 1, \quad a_{1} = a, \quad a_{n+1} = \left(\frac{a_{n}^{2}}{a_{n-1}^{2}} - 2\right) \cdot a_{n} .
$$
Show that for all integers $k \geq 0$, we have
$$
\sum_{i=0}^{k} \frac{1}{a_{i}} \lt \frac{a+2-\sqrt{a^{2}-4}}{2} .
$$