Relation of terms from sum ratios

If the sum of $m$ terms of an arithmetic progression is to the sum of $n$ terms as $m^2$ is to $n^2$, show that the $m^{\text{th}}$ term is to the $n^{\text{th}}$ term as $2m-1$ is to $2n-1$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. IV. Arithmetical Progression, Examples IV.b, No. 18