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.

Assume $x^2<n$ but $(x+1)^2\ge n$.  
Then, if $x\ge3, 2x+1<x^2,$ so $(x+1)^2=x^2+2x+1<2x^2<2n.$
The cases where $x\lt3$ can be easily eliminated.
