Question 18
Let $(a_n)_{n \ge 1}$ be a sequence of real numbers such that $a_1 \gt 2$ and $a_{n+1} = 1 + \frac{2}{a_n}$ for $n \ge 1$.
a) Prove that $a_{2n-1} + a_{2n} \gt 4$ for all $n \ge 1$ and $\lim_{n \to \infty} a_n = 2$.
b) Determine the largest real $a$ for which the inequality
$$
\sqrt{x^2 + a_1^2} + \sqrt{x^2 + a_2^2} + \sqrt{x^2 + a_3^2} + \dots + \sqrt{x^2 + a_n^2} \gt n\sqrt{x^2 + a^2}
$$
occurs for all $x \in \mathbb{R}$ and for any $n \in \mathbb{N}^*$.
a) Prove that $a_{2n-1} + a_{2n} \gt 4$ for all $n \ge 1$ and $\lim_{n \to \infty} a_n = 2$.
b) Determine the largest real $a$ for which the inequality
$$
\sqrt{x^2 + a_1^2} + \sqrt{x^2 + a_2^2} + \sqrt{x^2 + a_3^2} + \dots + \sqrt{x^2 + a_n^2} \gt n\sqrt{x^2 + a^2}
$$
occurs for all $x \in \mathbb{R}$ and for any $n \in \mathbb{N}^*$.