Non linear recursive sequence bounds: $14\le R_{100} \le 18$. Let $R_{n+1}=R_n+\frac{1}{R_n}$, where $R_1=1$.
I need to prove that $14\le R_{100} \le 18$.
Can anyone help please? 

Since $$R_n^2=R_{n-1}^2+\frac{1}{R_{n-1}^2}+2,$$ we obtain
$$R_{100}^2=R_1^2+\frac{1}{R_{99}^2}+...+\frac{1}{R_1^2}+2\cdot99>199>14^2.$$
Since, $R_{10}^2>2\cdot9+1=19,$ we obtain
$$R_{100}^2=\frac{1}{R_{99}^2}+...+\frac{1}{R_1^2}+199<\frac{89}{R_{10}^2}+\frac{10}{R_1^2}+199=\frac{89}{19}+10+199<324.$$
