Arithmetic sequences have a common difference between terms \[t_{n}=t_{n-1}+d\]

Geometric sequences have a common ratio \[t_{n}=t_{n-1}r\]

Harmonic sequences have the defining property that the reciprocals of their terms have a common difference (i.e. their reciprocals form an arithmetic sequence).

\[\frac{1}{t_{n}}=\frac{1}{t_{n-1}} +d\]

or

\[t_{n}=\frac{t_{n-1}}{1+ dt_{n-1}}\]

When you set the first term and the difference to 1, you get a harmonic sequence that has come to be known as

*the*harmonic sequence, namely:

\[1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \]

If

*a*,

*b*,

*c*are three three terms of an arithmetic sequence, the middle term

*b*is the 'arithmetic mean' of

*a*and

*c*, which is generally what we mean by "the mean" or average of

*a*and

*c*: \[b = \frac{a+c}{2} \] Similarly, if

*a*,

*b*,

*c*are three terms of a geometric sequence, then

*b*is the 'geometric mean' of

*a*and

*c*: \[b = \sqrt{ac} \] Not too surprisingly then, if

*a*,

*b*, and

*c*form a harmonic sequence, then

*b*is the 'harmonic mean' of

*a*and

*c:*\[b = \frac{2ac}{a+c} \]

What might be surprising at first is that if you have two numbers, say

*a*and

*c*, then the harmonic, geometric and arithmetic means of

*a*and

*c*form a geometric series (or put another way, the geometric mean of

*a*and

*c*is also the geometric mean of the harmonic and arithmetic means of

*a*and

*c*).

The algebra textbooks that I looked at did not present any applications of harmonic means or sequences (the exercises were restricted to formula manipulation), and the harmonic sequence does not seem to provide a very applicable 'model of growth', which is how the other sequences are generally presented. Harmonic relationships come up frequently in geometry, however, and one text did feature a nice construction for all three means (see below).

In this construction (GSP file here),

**O**is the center of the circle, and

**PR**is a secant that goes through

**O**.

**SP**is tangent to the circle, and

**ST**is at right angles to the secant

**PR**. If we let

*a*=

**PQ**and

*b*=

**PR,**

**PO**gives the arithmetic mean of

*a*and

*b*,

**PS**gives their geometric mean and

**PT**gives their harmonic mean. (Proof left as an exercise :)

I couldn't locate the texts I consulted in Google books, but for some other examples of how harmonic sequences were presented, see these:

H. S. Hall, S. R. Knight, Higher algebra: a sequel to elementary algebra for schools. Macmillan, 1894. (page 47)

O. Lodge, Easy mathematics, chiefly arithmetic..., Macmillan, 1906. (page 339)