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harmonic mean∗ drini† 2013-03-21 12:34:09 If a1 , a2 , . . . , an are positive numbers, we define their harmonic mean as the inverse number of the arithmetic mean of their inverse numbers: H.M. = 1 a1 + 1 a2 n + ··· + 1 an • It follows easily the estimation H.M. < nai (i = 1, 2, . . . , n). • If you travel from city A to city B at x miles per hour, and then you travel back at y miles per hour. What was the average velocity for the whole trip? The harmonic mean of x and y. That is, the average velocity is 1 x 2 + 1 y = 2xy . x+y • If one draws through the intersecting point of the diagonals of a trapezoid a line parallel to the parallel sides of the trapezoid, then the segment of the line inside the trapezoid is equal to the harmonic mean of the parallel sides. • In the harmonic series 1 1 1 + + + ... 2 3 4 every term equals to the harmonic mean of the term preceding it and the term following it. 1+ ∗ hHarmonicMeani created: h2013-03-21i by: hdrinii version: h30408i Privacy setting: h1i hDefinitioni h11-00i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1