Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.5 National olympiad Prove it

Example 1 Let the sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}$ satisfy $a_{0}=\frac{1}{2}$ , and $a_{k+1}=a_{k}+$ $\frac{1}{n} a_{k}^{2}, k=0,1,2, \cdots, n-1$ , prove that: $1-\frac{1}{n}<a_{n}<1$ . (1980 Mathematics Olympiad of Finland, UK, Germany, Hungary, and Sweden)

Solution

Proof: By induction, it is easy to see that $a_{0}\frac{1}{a_{n-1}}-\frac{1}{n}>\frac{1}{a_{n-2}}-\frac{2}{n}>\cdots>\frac{1}{a_{0}}-1=1$ , i.e., $a_{n}a_{k}\left(1+\frac{a_{k+1}}{n+1}\right)$ . Therefore, we have
$\frac{1}{a_{k+1}}\frac{n+1}{n+2}=1-\frac{1}{n+2}>1-\frac{1}{n}$ .
In summary, $1-\frac{1}{n}<a_{n}<1$ .

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