Harmonic progression (mathematics)

The first ten members of the harmonic sequence ${\displaystyle a_{n}={\tfrac {1}{n}}}$.

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

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

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

• Geometric progression
• Harmonic series

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

• 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