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单词 HarmonicMean
释义

harmonic mean


If  a1,a2,,an  are positive numbers, we define their harmonic meanMathworldPlanetmath as the inverse number of the arithmetic meanMathworldPlanetmath of their inverse numbers:

H.M.=n1a1+1a2++1an
  • 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

    21x+1y=2xyx+y.
  • If one draws through the intersecting point of the diagonals of a trapezoidMathworldPlanetmath a line parallelMathworldPlanetmathPlanetmath 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+12+13+14+

    every following it.

Titleharmonic mean
Canonical nameHarmonicMean
Date of creation2013-03-22 11:50:50
Last modified on2013-03-22 11:50:50
Ownerdrini (3)
Last modified bydrini (3)
Numerical id15
Authordrini (3)
Entry typeDefinition
Classificationmsc 11-00
Related topicArithmeticMean
Related topicGeneralMeansInequality
Related topicWeightedPowerMean
Related topicPowerMean
Related topicArithmeticGeometricMeansInequality
Related topicRootMeanSquare3
Related topicProofOfGeneralMeansInequality
Related topicProofOfArithmeticGeometricHarmonicMeansInequality
Related topicHarmonicMeanInTrapezoid
Related topicContraharmonicMean
Related topicContr
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更新时间:2025/5/4 23:43:12