Bounds for harmonic means

Let $p$ be the first of $n$ arithmetic means between two numbers $a$ and $b$, and let $q$ be the first of $n$ harmonic means between the same two numbers. Prove that $q$ cannot lie between $p$ and ${\left(\frac{n+1}{n-1}\right)}^{2}p$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VI. Harmonical Progression, Examples VI.a, No. 21