Relation of first term to difference

If $l$, $m$, $n$ are three numbers in geometric progression, prove that the first term of an arithmetic progression whose $l^{\text{th}}$, $m^{\text{th}}$, and $n^{\text{th}}$ terms are in harmonic progression is to the common difference as $m+1$ is to $1$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VI. Harmonical Progression, Examples VI.a, No. 17