In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an

arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

. 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 :

\frac,\ \frac,\ \frac,\ \frac, \cdots,

where ''a'' is not zero and −''a''/''d'' is not a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

, or a finite sequence of the form :

\frac,\ \frac,\ \frac,\ \frac, \cdots,\ \frac,

where ''a'' is not zero, ''k'' is a natural number and −''a''/''d'' is not a

or is greater than ''k''.

Examples

* 1, 1/2, 1/3, 1/4, 1/5, 1/6, sometimes referred to as the ''harmonic sequence'' * 12, 6, 4, 3,

\tfrac

, 2, … ,

\tfrac

, … * 30, −30, −10, −6, −

\tfrac

, … ,

\tfrac

* 10, 30, −30, −10, −6, −

, … ,

\tfrac

Sums of harmonic progressions

Infinite harmonic progressions are not summable (sum to infinity). It is not possible for a harmonic progression of distinct

unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...

s (other than the trivial case where ''a'' = 1 and ''k'' = 0) to sum to an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

. The reason is that, necessarily, at least one denominator of the progression will be

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

by a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that does not divide any other denominator.

Use in geometry

collinear points In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

A, B, C, and D are such that D is the

harmonic conjugate In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of ...

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.
Modern geometry of the point, straight line, and circle: an elementary treatise
' by

John Alexander Third John Alexander Third (1865–1948) was a Scottish mathematician. Life and work Third, son of a stonemason, was educated at Robert Gordon's College before entering in 1885 in the University of Aberdeen where he graduated D.Sc in 1889, after ...

(1898) p. 44 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 Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

are in

, then the sides are in harmonic progression.

Leaning Tower of Lire

An excellent example of Harmonic Progression is the

Leaning Tower of Lire In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire , also the book-stacking problem, or a number of other similar terms) is a puzzle concerning the stacking of blocks at the edge of a table. Statement The block ...

. In it, uniform blocks are stacked on top of each other to achieve the maximum sideways or lateral distance covered. The blocks are stacked 1/2, 1/4, 1/6, 1/8, 1/10, … distance sideways below the original block. This ensures that the center of gravity is just at the center of the structure so that it does not collapse. A slight increase in weight on the structure causes it to become unstable and fall.

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 Webster Wells (1851–1916) was an American mathematician known primarily for his authorship of mathematical textbooks. Early life and career Webster Wells was born at Roxbury, Massachusetts on September 4, 1851. His parents, Thomas Foster Well ...

(1897) p. 307 {{Series (mathematics) Mathematical series Sequences and series

Examples

Sums of harmonic progressions

Use in geometry

Leaning Tower of Lire

See also

References