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Difference between revisions of "Harmonic sequence"

(Created page with "In algebra, a '''harmonic sequence''', sometimes called a '''harmonic progression''', is a sequence of numbers such that the difference between the reciprocals of any...")
 
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More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is a harmonic progression if and only if <math>\frac{1}{a_2} - \frac{1}{a_1} = \frac{1}{a_3} - \frac{1}{a_2} = \cdots = \frac{1}{a_n} - \frac{1}{a_{n-1}}.</math> A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in harmonic progression if and only if <math>\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}</math>.
 
More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is a harmonic progression if and only if <math>\frac{1}{a_2} - \frac{1}{a_1} = \frac{1}{a_3} - \frac{1}{a_2} = \cdots = \frac{1}{a_n} - \frac{1}{a_{n-1}}.</math> A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in harmonic progression if and only if <math>\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}</math>.
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Revision as of 12:44, 24 November 2021

In algebra, a harmonic sequence, sometimes called a harmonic progression, is a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence.

For example, $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{99}, \frac{1}{91}, \frac{1}{83}, \frac{1}{75}$ are harmonic sequences; however, $1, 1, \frac{1}{3}, \frac{1}{5}$ and $\frac{1}{4}, \frac{1}{12}, \frac{1}{36}, \frac{1}{108}, \ldots$ are not.

More formally, the sequence $a_1, a_2, \ldots , a_n$ is a harmonic progression if and only if $\frac{1}{a_2} - \frac{1}{a_1} = \frac{1}{a_3} - \frac{1}{a_2} = \cdots = \frac{1}{a_n} - \frac{1}{a_{n-1}}.$ A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants $a$, $b$, and $c$ are in harmonic progression if and only if $\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}$.

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