An arithmetic progression, arithmetic sequence or linear sequence 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 ${\displaystyle a_{1}}$ and the common difference of successive members is ${\displaystyle d}$, then the ${\displaystyle n}$-th term of the sequence (${\displaystyle a_{n}}$) 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.

According to an anecdote of uncertain reliability, in primary school Carl Friedrich Gauss reinvented the formula ${\displaystyle {\tfrac {n(n+1)}{2}}}$ for summing the integers from 1 through ${\displaystyle n}$, for the case ${\displaystyle n=100}$, 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; in China to Zhang Qiujian; in India to Aryabhata, Brahmagupta and Bhaskara II; and in medieval Europe to Alcuin, Dicuil, Fibonacci, Sacrobosco, and anonymous commentators of Talmud known as Tosafists. Some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.

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 ${\displaystyle a_{1}}$ and ending with ${\displaystyle a_{n}}$. For example,