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 term equals to the harmonic mean of the term preceding it and the term following it.