Wednesday, September 9, 2009

a neglected sequence

Looking at some old texts (from the 1930s, mostly) I came across a type of sequence that was once part of the standard curriculum along side the familiar arithmetic and geometric varieties.  As far as I know, this third sequence type, harmonic sequences, is no longer part of any standard high school curriculum (please correct me if I am wrong).

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