Arithmetic progression

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

a

1

{\displaystyle a_{1}}

and the common difference of successive members is

d

{\displaystyle d}

, then the

n

{\displaystyle n}

-th term of the sequence (

a

n

{\displaystyle a_{n}}

) is given by

a

n

=

a

1

+

(

n

−

1

)

d

.

{\displaystyle a_{n}=a_{1}+(n-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.

History

According to an anecdote of uncertain reliability, in primary school Carl Friedrich Gauss reinvented the formula

n

(

n

+

1

)

2

{\displaystyle {\tfrac {n(n+1)}{2}}}

for summing the integers from 1 through

n

{\displaystyle n}

, for the case

n

=

100

{\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.