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{{For|the musical term|Chord progression}}
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

:<math> 1/a ,\ \frac{1}{a+d}\ , \frac{1}{a+2d}\ , \frac{1}{a+3d}\ , \cdots, \frac{1}{a+kd},</math>

where &minus;1/''d'' is not a [[natural number]] and ''k'' '''is''' a natural number.

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

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

==Examples==
:12, 6, 4, 3, <math>\tfrac{12}{5}</math>, 2, … , <math>\tfrac{12}{1+n}</math>

:10, 30, &minus;30, &minus;10, &minus;6, &minus; <math>\tfrac{30}{7}</math>, … , <math>\tfrac{10}{1-\tfrac{2n}{3}}</math>

==Use in geometry==
If [[Line (geometry)|collinear points]] A, B, C, and D are such that D is the [[Projective harmonic conjugates|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.<ref>''Chapters on the modern geometry of the point, line, and circle, Vol. II'' by Richard Townsend (1865) p. 24</ref><ref>''Modern geometry of the point, straight line, and circle: an elementary treatise'' by John Alexander Third (1898) p. 44</ref> Specifically, each of the sequences
AC,&nbsp;AB,&nbsp;AD; BC,&nbsp;BA,&nbsp;BD; CA,&nbsp;CD,&nbsp;CB; and DA,&nbsp;DC,&nbsp;DB are harmonic progressions, where each of the distances is signed according to a fixed orientation of the line.

==See also==
*[[Geometric progression]]
*[[Harmonic series (mathematics)|Harmonic series]]

==References==
{{reflist}}

*''Mastering Technical Mathematics'' by Stan Gibilisco, Norman H. Crowhurst, (2007) p.&nbsp;221
*''Standard mathematical tables'' by Chemical Rubber Company (1974) p.&nbsp;102
*''Essentials of algebra for secondary schools'' by [[Webster Wells]] (1897) p.&nbsp;307

[[Category:Mathematical series]]
[[Category:Sequences and series]]

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