#1064 · Algebra · Sequences & series

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 $(\frac{n + 1}{n - 1})^{2} p$ .

~12 minProof · checked by AIarithmetic meansharmonic meansinequalities

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

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