Harmonic progression (mathematics)

For the musical term, see 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

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

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.

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.

See also

• Geometric progression
• Harmonic series

References

• 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