Work Step by Step
1 + $(2n-1)$ is $2n$. So is $3 + (2n - 3)$. And so on. This occurs with each of the $n/2$ pairings of numbers. Thus after rearranging, the sum $1 + 3 + 5 + \cdots + (2n - 1)$ is $2n\cdot (n/2) = n^2$. This can be proved more formally by induction.