Competition problem — Number theory

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Number theory Difficulty 6.1 National olympiad Prove it

Example 9 Arrange a strip of width 1 using rectangles of width 1 and lengths $a_{1}, a_{2}, a_{3}, \cdots \left(a_{1} \neq 1\right)$ , which are placed tightly and without overlapping. Starting from the second rectangle, each rectangle is similar but not congruent to the rectangle formed by the previously arranged rectangles. After placing the first $n$ rectangles, the length of the strip covered is denoted as $S_{n}$ . Does there exist a real number that is not exceeded by $S_{n}$ ? Prove your conclusion.

Solution

Proof As shown in Figure $13-3$ , according to the problem, we have $1: S_{n}=a_{n+1}: 1$ , so $a_{n+1}=\frac{1}{S_{n}}$ . Since $S_{n+1}=S_{n}+a_{n+1}$ , it follows that for all $n(n \in \mathbf{N})$ , we have $S_{n+1}=S_{n}+\frac{1}{S_{n}}$ .
Therefore, $S_{n+1}^{2}=\left(S_{n}+\frac{1}{S_{n}}\right)^{2}>S_{n}^{2}+2$
which means $S_{n+1}^{2}-S_{n}^{2}>2 \quad n=1,2, \cdots$
Summing up, we get $S_{n+1}^{2}-S_{1}^{2}>2 n$ , i.e., $S_{n+1}^{2}>2 n+S_{1}^{2}>2 n$

Thus, the sequence $\left\{S_{n}^{2}\right\}$ is monotonically increasing and unbounded, and so is $\left\{S_{n}\right\}$ . Therefore, there does not exist a real number that is not exceeded by $S_{n}$ .

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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.