Download PDF

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