Prove for any integer $n > 2$, one can find n distinct positive integers, such that the sum of their reciprocals is equal to 1.
Is there any non-complicated way to do this? Induction doesn't seem to work, and neither does proof by contradiction. Writing out the first couple of $n$ hasn't seemed to lead me anywhere.
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1$\begingroup$ Titles are meant to be just that: titles. The question should be located in the body along with some of your own thoughts...such as why induction failed or where you get stuck in a proof by contradiction. $\endgroup$– ClaytonCommented Oct 31, 2017 at 0:22
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1 Answer
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$$ \frac{1}{k} = \frac{1}{k+1} + \frac{1}{k^2 + k} $$
answered Oct 31, 2017 at 0:39
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$\begingroup$ Yes, but this doesn't take into account that you need exactly n integers' reciprocals. $\endgroup$ Commented Nov 3, 2017 at 23:55
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$\begingroup$ @GerardL. actually, it does. How about if you begin with $1/2, 1/3, 1/6$ and see how to get four terms. $\endgroup$ Commented Nov 4, 2017 at 0:16
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$\begingroup$ I don't see your answer's application. $\endgroup$ Commented Nov 4, 2017 at 0:30