# harmonic mean

If  $a_{1},\,a_{2},\,\ldots,\,a_{n}$  are positive numbers, we define their harmonic mean as the inverse number of the arithmetic mean of their inverse numbers:

 $H.M.\;=\;\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}}$
• It follows easily the estimation

 $H.M.\;<\;na_{i}\quad(i\;=\;1,\,2,\,\ldots,\,n).$
• 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

 $\frac{2}{\frac{1}{x}+\frac{1}{y}}\;=\;\frac{2xy}{x\!+\!y}.$
• 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

 $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...$

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