Track / Stage 5 / 305 of 400 #905 of 1964

Problem 905

AIME late

Algebra Difficulty 5.7 Prove it

8. Given $a_{1}=1, a_{n+1}=\frac{a_{n}}{n}+\frac{n}{a}$ . Prove that for $n \geqslant 4$ , $\sqrt{n}<a_{n}<\sqrt{n+1}$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

8. First prove that $f(x)=\frac{x}{n}+\frac{n}{x}$ is a decreasing function on $(0, n)$ , then use mathematical induction to prove $\sqrt{n} \leqslant a_{n} \leqslant \frac{n}{\sqrt{n-1}}$ , and then prove $a_{n}<\sqrt{n+1}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.