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Problem

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Tags: algebra, equation, egyptian fractions

kapsitis

23.07.2016 00:04

Prove that for every integer $n$ ( $n > 1$ ) there exist two positive integers $x$ and $y$ ( $x \leq y$ ) such that $\frac{1}{n} = \frac{1}{x(x+1)} + \frac{1}{(x+1)(x+2)} + \cdots + \frac{1}{y(y+1)}$

fractals

23.07.2016 00:07

$x = n - 1, y = n^2 - n - 1$ . Then

$\frac{1}{x(x + 1)} + \cdots + \frac{1}{y(y + 1)} = \frac{1}{x} - \frac{1}{y + 1} = \frac{1}{n - 1} - \frac{1}{n^2 - n} = \frac{1}{n}$ and we are done, since

$0 < x \le y$ is clear from

$n \ge 2$ .