Arithmetic progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence.
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
) is given by A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.
According to an anecdote of uncertain reliability,[1] in primary school Carl Friedrich Gauss reinvented the formula
, by grouping the numbers from both ends of the sequence into pairs summing to 101 and multiplying by the number of pairs.
Regardless of the truth of this story, Gauss was not the first to discover this formula.
Similar rules were known in antiquity to Archimedes, Hypsicles and Diophantus;[2] in China to Zhang Qiujian; in India to Aryabhata, Brahmagupta and Bhaskara II;[3] and in medieval Europe to Alcuin,[4] Dicuil,[5] Fibonacci,[6] Sacrobosco,[7] and anonymous commentators of Talmud known as Tosafists.
[8] Some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.
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).
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 arithmetic progression of real numbers beginning with
For example, 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 essentially the same as the formula for the mean of a discrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.
The product of the members of a finite arithmetic progression with an initial element a1, common differences d, and n elements in total is determined in a closed expression where
denotes the Gamma function.
This is a generalization of the facts that the product of the progression
and that the product for positive integers
denotes the rising factorial.
, the product of the terms of the arithmetic progression given by
up to the 50th term is The product of the first 10 odd numbers
is given by The standard deviation of any arithmetic progression is where
The formula is essentially the same as the formula for the standard deviation of a discrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.
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.
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.
denote the number of arithmetic subsets of length
ϕ ( η , κ )
ϕ ( η , κ ) =
arithmetic subsets and, counting directly, one sees that there are 9; these are
