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
and the common difference of successive members is
, then the
-th term of the sequence (
) is given by:
:
,
If there are ''m'' terms in the AP, then
represents the last term which is given by:
:
.
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:
:
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:
:
In the case above, this gives the equation:
:
This formula works for any real numbers
and
. For example: this
:
Derivation
To derive the above formula, begin by expressing the arithmetic series in two different ways:
:
:
Rewriting the terms in reverse order:
:
Adding the corresponding terms of both sides of the two equations and halving both sides:
:
This formula can be simplified as:
:
Furthermore, the mean value of the series can be calculated via:
:
:
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
:
where
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
is negative or zero.
This is a generalization from the fact that the product of the progression
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 ...
and that the product
:
for
positive integers
and
is given by
:
Derivation
:
where
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
, valid for a complex number
,
:
,
:
,
so that
:
for
a positive integer and
a positive complex number.
Thus, if
,
:
,
and, finally,
:
Examples
;Example 1
Taking the example
, the product of the terms of the arithmetic progression given by
up to the 50th term is
:
; Example 2
The product of the first 10 odd numbers
is given by
:
=
Standard deviation
The standard deviation of any arithmetic progression can be calculated as
:
where
is the number of terms in the progression and
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 ...
*
References
External links
*
*
*
{{DEFAULTSORT:Arithmetic Progression
Arithmetic series
Articles containing proofs
Sequences and series