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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 −1/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.

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. 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.

Geometric progression
Harmonic series

References

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", Erdös 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.
Chapters on the modern geometry of the point, line, and circle, Vol. II by Richard Townsend (1865) p. 24

Modern geometry of the point, straight line, and circle: an elementary treatise by John Alexander Third (1898) p. 44

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