Relation for equal means

Find the relation between $x$ and $y$ so that the $r^{\text{th}}$ arithmetic mean between $x$ and $2y$, with $n$ means inserted, is the same as the $r^{\text{th}}$ mean between $2x$ and $y$, also with $n$ means inserted.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. IV. Arithmetical Progression, Examples IV.b, No. 23