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.

Question

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.

Answer 1

$$ \frac{1}{k} = \frac{1}{k+1} + \frac{1}{k^2 + k} $$