Harmonic progression (mathematics)

For the musical term, see Chord progression.
The first ten members of the harmonic sequence a_n=\tfrac 1n.

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. In other words, it is a sequence of the form

1/a ,\ \frac{1}{a+d}\ , \frac{1}{a+2d}\ , \frac{1}{a+3d}\ , \cdots, \frac{1}{a+kd},

where −a/d is not a natural number and k is a natural number.

(Terms in the form \frac{x}{y+z}\ can be expressed as \frac{\frac{x}{y}}{\frac{y+z}{y}} , we can let \frac{x}{y}=a and \frac{z}{y}=kd .)

Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.[1]

Examples

12, 6, 4, 3, \tfrac{12}{5}, 2, … , \tfrac{12}{1+n}
10, 30, −30, −10, −6, − \tfrac{30}{7}, … , \tfrac{10}{1-\tfrac{2n}{3}}

Use in geometry

If collinear points A, B, C, and D are such that D is the harmonic conjugate of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression.[2][3] Specifically, each of the sequences AC, AB, AD; BC, BA, BD; CA, CD, CB; and DA, DC, DB are harmonic progressions, where each of the distances is signed according to a fixed orientation of the line.

See also

References

  1. ↑ Erdős, P. (1932), "Egy Kürschák-féle elemi számelméleti tétel általánosítása" [Generalization of an elementary number-theoretic theorem of Kürschák] (PDF), Mat. Fiz. Lapok (in Hungarian) 39: 17–24. As cited by Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions", Erdős centennial, Bolyai Soc. Math. Stud. 25, János Bolyai Math. Soc., Budapest, pp. 289–309, doi:10.1007/978-3-642-39286-3_9, MR 3203600.
  2. ↑ Chapters on the modern geometry of the point, line, and circle, Vol. II by Richard Townsend (1865) p. 24
  3. ↑ Modern geometry of the point, straight line, and circle: an elementary treatise by John Alexander Third (1898) p. 44
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