Harmonic progression (mathematics)

Last updated April 29, 2021

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

Examples

1, 1/2, 1/3, 1/4, 1/5, 1/6, sometimes referred to as the harmonic sequence
12, 6, 4, 3, ${\tfrac {12}{5}}$ , 2, … , ${\tfrac {12}{n}}$ , …
30, −30, −10, −6, − ${\tfrac {30}{7}}$ , … , ${\tfrac {10}{1-{\tfrac {2n}{3}}}}$
10, 30, −30, −10, −6, − , … , ${\tfrac {10}{1-{\tfrac {2(n-1)}{3}}}}$

Sums of harmonic progressions

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]

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.

Leaning Tower of Lire

An excellent example of Harmonic Progression is the Leaning Tower of Lire. 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.

Related Research Articles

In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers $x$ and $y$ is defined as follows:

In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

There are several kinds of mean in mathematics, especially in statistics:

Eulers totient function Gives the number of integers relatively prime to its input

In number theory, Euler's totient function counts the positive integers up to a given integer $n$ that are relatively prime to $n$ . It is written using the Greek letter phi as $or, and may also be called Euler's phi function . In other words, it is the number of integers k in the range 1 \leq k \leq n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n .$

In mathematics, the harmonic series is the divergent infinite series

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The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

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Harmonic number Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n

In mathematics, the $n$ -th harmonic number is the sum of the reciprocals of the first $n$ natural numbers:

$Egyptian fraction Finite sum of distinct unit fractions$

An Egyptian fraction is a finite sum of distinct unit fractions, such as

The square root of 2, or the one-half power of 2, written in mathematics as $or, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the principal square root of 2, to distinguish it from the negative number with the same property.$

A powerful number is a positive integer m such that for every prime number p dividing m, p² also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a²b³, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

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.

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of a numerator displayed above a line, and a non-zero denominator, displayed below that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

In combinatorial mathematics, a large set of positive integers

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

In mathematics, an arithmetico–geometric sequence is the result of the term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put more plainly, the nth term of an arithmetico–geometric sequence is the product of the nth term of an arithmetic sequence and the nth term of a geometric one. Arithmetico–geometric sequences arise in various applications, such as the computation of expected values in probability theory. For instance, the sequence

Ganita Kaumudi is a treatise on mathematics written by Indian mathematician Narayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa" by Narayana Pandit. It was written as a commentary on the Līlāvatī by Bhāskara II.

References

↑ Erdős, P. (1932), "Egy Kürschák-féle elemi számelméleti tétel általánosítása" [Generalization of an elementary number-theoretic theorem of Kürschák](PDF), Mat. Fiz. Lapok (in Hungarian), 39: 17–24CS1 maint: discouraged parameter (link). As cited by Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions", Erdős centennial, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, pp. 289–309, CiteSeerX 10.1.1.300.91 , doi:10.1007/978-3-642-39286-3_9, ISBN 978-3-642-39285-6, MR 3203600 CS1 maint: discouraged parameter (link).
↑ Chapters on the modern geometry of the point, line, and circle, Vol. II by Richard Townsend (1865) p. 24
↑ Modern geometry of the point, straight line, and circle: an elementary treatise by John Alexander Third (1898) p. 44

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 (1897) p. 307

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[1] Erdős, P. (1932), "Egy Kürschák-féle elemi számelméleti tétel általánosítása" [Generalization of an elementary number-theoretic theorem of Kürschák](PDF), Mat. Fiz. Lapok (in Hungarian), 39: 17–24CS1 maint: discouraged parameter (link). As cited by Graham, Ronald L. (2013), "Paul Erdős and Egyptian fractions", Erdős centennial, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, pp. 289–309, CiteSeerX 10.1.1.300.91 , doi:10.1007/978-3-642-39286-3_9, ISBN 978-3-642-39285-6, MR 3203600 CS1 maint: discouraged parameter (link).

[2] Chapters on the modern geometry of the point, line, and circle, Vol. II by Richard Townsend (1865) p. 24

[3] Modern geometry of the point, straight line, and circle: an elementary treatise by John Alexander Third (1898) p. 44

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