Ratio of difference to first term

If the $(m+1{)}^{\text{th}}$, $(n+1{)}^{\text{th}}$, and $(r+1{)}^{\text{th}}$ terms of an arithmetic progression are in geometric progression, and $m$, $n$, $r$ are in harmonic progression, show that the ratio of the common difference to the first term of the arithmetic progression is $-\frac{2}{n}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VI. Harmonical Progression, Examples VI.a, No. 16