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)

1 comment:

  1. One of the oldest types of math problems in existence *from the Rhind Papyrus)is of the harmonic mean type see


    and they still show up in problems about crossed ladders and the average speed from here to there if you drive 50 miles/hour there and 30 miles/hour back.. and in probability they play a part in helping figure out how long it will be until the next record breaking event...