# 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. 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.

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

## 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.

In a triangle, if the altitudes are in arithmetic progression, then the sides are in harmonic progression