Prove that $A_{100} \gt 14$ where $A_{n}=A_{n-1}+\frac{1}{A_{n-1}}$ and $A_1=1$

Question

I tried attempting the question, and the best upper bound I could obtain was $1+\ln{98}$. I tried using $A_{n}\le n$ to form a harmonic series, but that wasn't strong enough. Any help would be appreciated, Thanks.

Answer 1

HINT: Try squaring your recursion.

Answer 2

Just to spell out Michael's answer:

Let $B_n = A_n^2 + \frac{1}{A_n^2}$. Squaring the recursive relation, you find $B_n > A_n^2 = B_{n-1} + 2$. Hence, $B_{100} > B_{99} + 2 > \dots > B_1 + 198 = 200$. It follows that $A_{100}^2 + \frac{1}{A_{100}^2} > 200$. Solving, as you can, equation $x^2 + \frac{1}{x^2} = 200$, and using that $A_{100} > 1$, you find that $$A_{100} > \sqrt{100+3 \sqrt{1111}} = 14.142\dots > 14,$$ which is what you needed.

Answer 3

Another fleshing out of Michael's answer:

Let $B_n = A_n^2$. Then $B_n=A_{n-1}^2 + 2 + \frac{1}{A_{n-1}^2} > B_{n-1} + 2$.

Thus, $B_n > B_1 + (n-1)2=2n-1$.

Now, $A_n > 14$ iff $B_n > 196$, and so it suffices to take $2n-1>196$, that is, $n \ge 99$.