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An arithmetic progression or arithmetic sequence () is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s 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 difference of 2. If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the n-th term of the sequence (a_n) is given by: :a_n = a + (n - 1)d, If there are ''m'' terms in the AP, then a_m represents the last term which is given by: :a_m = a + (m - 1)d. A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.


Sum

Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum: :2 + 5 + 8 + 11 + 14 = 40 This sum can be found quickly by taking the number ''n'' of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2: :\frac In the case above, this gives the equation: :2 + 5 + 8 + 11 + 14 = \frac = \frac = 40. This formula works for any real numbers a_1 and a_n. For example: this :\left(-\frac\right) + \left(-\frac\right) + \frac = \frac = -\frac.


Derivation

To derive the above formula, begin by expressing the arithmetic series in two different ways: : S_n=a+a_2+a_3+\dots+a_ +a_n : S_n=a+(a+d)+(a+2d)+\dots+(a+(n-2)d)+(a+(n-1)d). Rewriting the terms in reverse order: : S_n=(a+(n-1)d)+(a+(n-2)d)+\dots+(a+2d)+(a+d)+a. Adding the corresponding terms of both sides of the two equations and halving both sides: : S_n=\frac a + (n-1)d This formula can be simplified as: :\begin S_n &=\frac + a + (n-1)d\\ &=\frac(a+a_n).\\ &=\frac(\text+\text). \end Furthermore, the mean value of the series can be calculated via: S_n / n: : \overline =\frac. The formula is very similar to the mean of a discrete uniform distribution.


Product

The
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of the members of a finite arithmetic progression with an initial element ''a''1, common differences ''d'', and ''n'' elements in total is determined in a closed expression :a_1a_2a_3\cdots a_n = a_1(a_1+d)(a_1+2d)...(a_1+(n-1)d)= \prod_^ (a_1+kd) = d^n \frac where \Gamma denotes the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. The formula is not valid when a_1/d is negative or zero. This is a generalization from the fact that the product of the progression 1 \times 2 \times \cdots \times n is given by the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
n! and that the product :m \times (m+1) \times (m+2) \times \cdots \times (n-2) \times (n-1) \times n for positive integers m and n is given by :\frac.


Derivation

:\begin a_1a_2a_3\cdots a_n &=\prod_^ (a_1+kd) \\ &= \prod_^ d\left(\frac+k\right) = d \left (\frac\right) d \left (\frac+1 \right )d \left ( \frac+2 \right )\cdots d \left ( \frac+(n-1) \right ) \\ &= d^n\prod_^ \left(\frac+k\right)=d^n ^ \end where x^ denotes the
rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
. By the recurrence formula \Gamma(z+1)=z\Gamma(z), valid for a complex number z>0, :\Gamma(z+2)=(z+1)\Gamma(z+1)=(z+1)z\Gamma(z), :\Gamma(z+3)=(z+2)\Gamma(z+2)=(z+2)(z+1)z\Gamma(z), so that : \frac = \prod_^(z+k) for m a positive integer and z a positive complex number. Thus, if a_1/d > 0 , :\prod_^ \left(\frac+k\right)= \frac, and, finally, :a_1a_2a_3\cdots a_n = d^n\prod_^ \left(\frac+k\right) = d^n \frac


Examples

;Example 1 Taking the example 3, 8, 13, 18, 23, 28, \ldots , the product of the terms of the arithmetic progression given by a_n = 3 + 5(n-1) up to the 50th term is :P_ = 5^ \cdot \frac \approx 3.78438 \times 10^. ; Example 2 The product of the first 10 odd numbers (1,3,5,7,9,11,13,15,17,19) is given by : 1.3.5\cdots 19 =\prod_^ (1+2k) = 2^ \cdot \frac =


Standard deviation

The standard deviation of any arithmetic progression can be calculated as : \sigma = , d, \sqrt where n is the number of terms in the progression and d is the common difference between terms. The formula is very similar to the standard deviation of a discrete uniform distribution.


Intersections

The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
. If each pair of progressions in a family of doubly infinite arithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is, infinite arithmetic progressions form a
Helly family In combinatorics, a Helly family of order is a family of sets in which every minimal ''subfamily with an empty intersection'' has or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non ...
. However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.


History

According to an anecdote of uncertain reliability, young
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in primary school reinvented this method to compute the sum of the integers from 1 through 100, by multiplying pairs of numbers in the sum by the values of each pair . However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
in the 5th century BC. Similar rules were known in antiquity to
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
,
Hypsicles Hypsicles ( grc-gre, Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancient Greek mathematician and astronomer known for authoring ''On Ascensions'' (Ἀναφορικός) and the Book XIV of Euclid's ''Elements''. Hypsicles lived in Alexandria. ...
and
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
; in China to Zhang Qiujian; in India to
Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
,
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
and Bhaskara II; and in medieval Europe to
Alcuin Alcuin of York (; la, Flaccus Albinus Alcuinus; 735 – 19 May 804) – also called Ealhwine, Alhwin, or Alchoin – was a scholar, clergyman, poet, and teacher from York, Northumbria. He was born around 735 and became the student o ...
,Problems to Sharpen the Young
John Hadley and David Singmaster, ''The Mathematical Gazette'', 76, #475 (March 1992), pp. 102–126.
Dicuil Dicuilus (or the more vernacular version of the name Dícuil) was an Irish monk and geographer, born during the second half of the 8th century. Background The exact dates of Dicuil's birth and death are unknown. Of his life nothing is known exce ...
,
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
,
Sacrobosco Johannes de Sacrobosco, also written Ioannes de Sacro Bosco, later called John of Holywood or John of Holybush ( 1195 – 1256), was a scholar, monk, and astronomer who taught at the University of Paris. He wrote a short introduction to the Hi ...
and to anonymous commentators of
Talmud The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the cente ...
known as
Tosafists Tosafists were rabbis of France and Germany, who lived from the 12th to the mid-15th centuries, in the period of Rishonim. The Tosafists composed critical and explanatory glosses (questions, notes, interpretations, rulings and sources) on the Tal ...
.Stern, M. (1990). 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. The Mathematical Gazette, 74(468), 157-159. doi:10.2307/3619368


See also

*
Geometric progression 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 ex ...
* Harmonic progression *
Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
*
Arithmetico-geometric sequence In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the ''n''th term of an arithmetico-geometric sequence ...
*
Inequality of arithmetic and geometric means In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
*
Primes in arithmetic progression In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n ...
*
Linear difference equation Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
*
Generalized arithmetic progression In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithmetic progression is generated by a single ...
, a set of integers constructed as an arithmetic progression is, but allowing several possible differences * Heronian triangles with sides in arithmetic progression *
Problems involving arithmetic progressions Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view. Largest progression-free subsets Find the cardinality (denoted by ''A'k''(''m' ...
*
Utonality ''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity), respectively. For example: , , ,... or , , ,.... Definition ...
*
Polynomials calculating sums of powers of arithmetic progressions The polynomials calculating sums of powers of arithmetic progressions are polynomials in a variable that depend both on the particular arithmetic progression constituting the basis of the summed powers and on the constant exponent, non-negative in ...


References


External links

* * * {{DEFAULTSORT:Arithmetic Progression Arithmetic series Articles containing proofs Sequences and series