Harmonic progression (mathematics)

In the following n is a natural number, in sequence: ${\displaystyle \ n=1,\ 2,\ 3,\ 4,\ \ldots \ }$

• ${\displaystyle 1,{\tfrac {\ 1\ }{2}},\ {\tfrac {\ 1\ }{3}},\ {\tfrac {\ 1\ }{4}},\ {\tfrac {\ 1\ }{5}},\ {\tfrac {\ 1\ }{6}},\ \ldots \ ,\ {\tfrac {\ 1\ }{n}},\ \ldots \ }$ is called the harmonic sequence
• 12, 6, 4, 3, ${\displaystyle \ {\tfrac {12}{\ 5\ }},\ 2,\ \ldots \ ,\ {\tfrac {12}{\ n\ }},\ \ldots \ }$
• 30, −30, −10, −6, ${\displaystyle \ -{\tfrac {30}{\ 7\ }},\ \ldots \ ,\ {\tfrac {30}{\ \left(3\ -\ 2n\right)\ }},\ \ldots \ }$
• 10, 30, −30, −10, −6, ${\displaystyle \ \ldots \ ,\ {\tfrac {30}{\ \left(5\ -\ 2n\right)\ }},\ \ldots \ }$

Infinite harmonic progressions are not summable (sum to infinity).

It is not possible for a harmonic progression of distinct unit fractions (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]

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.