Arithmetic progression
An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. 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:
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.
History[edit]
According to an anecdote of uncertain reliability,[1] young Carl Friedrich Gauss, who was in primary school, reinvented the formula for summing the integers from 1 through , for the case , by grouping the numbers from both ends of the sequence into pairs summing to 101 and multiplying by the number of pairs. 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 in the 5th century BC.[2] Similar rules were known in antiquity to Archimedes, Hypsicles and Diophantus;[3] in China to Zhang Qiujian; in India to Aryabhata, Brahmagupta and Bhaskara II;[4] and in medieval Europe to Alcuin,[5] Dicuil,[6] Fibonacci,[7] Sacrobosco[8] and to anonymous commentators of Talmud known as Tosafists.[9]
Intersections[edit]
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.[10] However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.