#1063 · Algebra · Sequences & series

Means ratio identity

Between any two positive quantities insert two arithmetic means $A_{1}, A_{2}$ , two geometric means $G_{1}, G_{2}$ , and two harmonic means $H_{1}, H_{2}$ . Show that $\frac{G _{1} G _{2}}{H _{1} H _{2}} = \frac{A _{1} + A _{2}}{H _{1} + H _{2}} .$

~8 minProof · checked by AImeansproportionality

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VI. Harmonical Progression, Examples VI.a, No. 20

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