Harmonic progression (mathematics)

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence.

Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

As a third equivalent characterization, it is an infinite sequence of the form

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

where a is not zero and −a/d is not a natural number, or a finite 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 is not zero, k is a natural number and −a/d is not a natural number or is greater than k.