# Prove that for any integer with $n>2$, I can find n distinct positive integers such that the sum of the reciprocals is equal to 1.

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.

• 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. – Clayton Oct 31 '17 at 0:22

$$\frac{1}{k} = \frac{1}{k+1} + \frac{1}{k^2 + k}$$
• @GerardL. actually, it does. How about if you begin with $1/2, 1/3, 1/6$ and see how to get four terms. – Will Jagy Nov 4 '17 at 0:16