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:

  • 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

  • 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


    every following it.

Title harmonic mean
Canonical name HarmonicMean
Date of creation 2013-03-22 11:50:50
Last modified on 2013-03-22 11:50:50
Owner drini (3)
Last modified by drini (3)
Numerical id 15
Author drini (3)
Entry type Definition
Classification msc 11-00
Related topic ArithmeticMean
Related topic GeneralMeansInequality
Related topic WeightedPowerMean
Related topic PowerMean
Related topic ArithmeticGeometricMeansInequality
Related topic RootMeanSquare3
Related topic ProofOfGeneralMeansInequality
Related topic ProofOfArithmeticGeometricHarmonicMeansInequality
Related topic HarmonicMeanInTrapezoid
Related topic ContraharmonicMean
Related topic Contr