Stephen R. WassellDepartment of Mathematical SciencesSweet Briar College Sweet Briar, Virginia USA
First we should agree on terminology and notation. An arithmetic
sequence can be defined as a never-ending list of (real) numbers
such that, taking any three in a row, the second is the arithmetic
mean of the first and third. An example would be {4,8,12,16,
...}. Using mathematical notation to generalize this, let { as is supposed to be the case. The situation is directly analogous for a geometric sequence,
whereby the second of any three numbers in a row must be the
geometric mean of the first and third. Using the same notation
as above, the requirement is that .
The general form also follows the above discussion, with addition
replaced by multiplication: { ar , so that^{n}as required. For examples, we can take We easily see that the list of numbers comprising an arithmetic
or a geometric sequence increases without bound. In the examples
we have been using positive numbers. We could consider negative
arithmetic or geometric sequences that We can follow the same setup with a harmonic sequence, requiring
that the second of any three numbers in a row is the harmonic
mean of the first and third, i.e., that
. The complication is that the general form for a harmonic sequence
is much less intuitive, and this is undoubtedly why Marcus posed
his question to me. My first instinct was to start with two (positive
real) numbers The formula for Solving this for For example, if we start with We can find a formula for the reverse direction of a bi-directional
harmonic sequence by finding Returning to the forward direction, we see that if at any point in the generation of a harmonic sequence, the last number is double the second-to-last number, we will get a zero denominator if we try to find the next number in the sequence. What if this is not the case? Experimentation seems to indicate the alternative is that at some point the last number will be greater than the second-to-last number, in which case the new denominator will be negative, and in fact the sequence will be negative from this point onwards![5] Here are two examples: . We see that the forward direction either repeats the same magnitudes as in the reverse direction (as in the second example) or takes a different path in approaching zero (as in the first example).[6] After fiddling with examples and noticing patterns, I discovered
that referring back to the general form of the arithmetic and
geometric sequences is most helpful.[7] We always start with two given numbers,
say Focusing on the forward direction of the sequence, it is easy
to see that which happens to be the same as in the arithmetic sequence. Continuing in this manner, the next three terms are In other words, the general form is which answers Marcus's question.[9] Because To prove the validity of the general form, suppose we are
given any harmonic sequence; we need to show that it can be realized
with our general form. Pick _{n}, a+1, comprise a harmonic triplet.[10] Naturally,
the reverse direction of a bi-directional harmonic sequence is
obtained by taking negative integer values for _{n}n.As for the claims about the behavior of harmonic sequences,
let us revisit the perspective whereby we start with
sum of terms,
whereas a sequence is an infinite list of terms (as is
a progression). return to text[2] Perhaps the most important classical use of geometric
sequences is in the "Pythagorean lambda", which "is
replete with arithmetic, geometric and harmonic means" [March
1998: 73]. See also [Wittkower 1998: 105-106; March 1998: 76,
96; Kappraff 2000: 48]. The Pythagorean lambda is based upon
the geometric sequences {1,2,4,8,...}and {1,3,9,27,...}. [3] It is interesting that for geometric sequences,
negative numbers are not as naturally unified with positive numbers,
as they are with the arithmetic sequences. [4]
In the concluding chapter of his highly influential treatise,
Nicomachus offers the proportion 6:8::9:12 as an example of "the
most perfect proportion" [Nicomachus 1926: 284-286; cf.
March 1998: 96]. [5]
Since we start with two positive numbers to begin the process,
the sequence will stay positive until this situation just described
occurs. [6]
In either case, it is interesting to take the view that +¥ and -¥ are one
and the same, as is sometimes done in 'completing' the real number
system by adding a single 'point' called, simply, ¥. With this
viewpoint, harmonic sequences that do not 'dead end' on ¥ instead
'pass through' ¥ and proceed to the negative side of the real number
line. After this, such harmonic sequences then increase to 0
just as the reverse direction decreases to 0. [7]
A search of the mathematical literature did not produce any ready-made
formulas for harmonic sequences; if the reader knows of a suitable
reference to this material, I would be grateful to be informed.
[8]
This process may make some intuitive sense if an alternate formulation
for the harmonic mean is used, namely that its reciprocal is
the arithmetic mean of the reciprocals of the extremes; in other
words, for [9]
A more attractive general formula may be obtained by setting
(subtracting return to text[10] Here it is easier to use the alternate formula for the harmonic mean described in footnote 8; in other words, check that return to text
March, Lionel. 1998. Nicomachus. 1926. Wittkower, Rudolf. 1998. Architectural Principles
in the Age of Humanism. Chichester, West Sussex: Academy Editions.
NNJ.
Copyright ©2005 Kim Williams Books top of
page |
NNJ HomepageGeometer's
Angle HomepageAbout
the AuthorComment on this articleRelated
Sites on the WWWOrder
books!Research
ArticlesDidacticsConference and Exhibit ReportsBook
ReviewsThe Virtual LibrarySubmission GuidelinesTop
of Page |