Non linear recursive sequence bounds: $14\le R_{100} \le 18$.

Question

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?

Answer 1

Let $S_n=R_n^2$. Then $$S_{n+1}=S_n+2+\frac1{S_n}.$$ So $S_n$ increases by at least $2$ and at most $3$ at each stage. As $S_1=1$ then what does this tell us about $S_{100}$?

Answer 2

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

$$R_{100}^2=\frac{1}{R_{99}^2}+...+\frac{1}{R_1^2}+199<\frac{89}{R_{10}^2}+\frac{10}{R_1^2}+199=\frac{89}{19}+10+199<324.$$