Prove for every positive integer $n$, one of $n, n+1, n+2,...,2n$ is the square of an integer Prove for every positive integer $n$, one of $n, n+1, n+2,...,2n$ is the square of an integer.
This seems like a proof by induction, but I'm more used to using proof by induction with a single equation, where I usually prove it's true for the base case then assume it's true for n and prove n+1 this is true, but I don't think that applies here.

The cases from $n=1$ to $n=6$ are clear. As for $n\geq 6$, since 
$$\sqrt{2n}-\sqrt{n}=(\sqrt{2}-1)\sqrt{n}\geq(\sqrt{2}-1)\sqrt{6}>1,$$
there will always be an integer $k$ between them for which $n\leq k^2\leq 2n$. 
P.S. This terminology does not look very satisfying, but I hope it would help after all. 
