# arithmetic sequence

An arithmetic sequence, also known as an **arithmetic progression**,
is a finite sequence of at least three numbers,
or an infinite sequence, whose terms differ by a constant, known as the **common difference**.

For example, starting with 1 and using a common difference of 4 we can get
the finite arithmetic sequence: 1, 5, 9, 13, 17, 21, and also the infinite
sequence 1, 5, 9, 13, 17, 21, 25, 29, ..., 4*n* + 1, ... In general,
the terms of an arithmetic sequence with the first term *a*_{0} and common difference *d*, have the form *a _{n}* =

*dn*+

*a*

_{0}(

*n*= 1, 2, 3,...).

Does every increasing sequence of integers have to contain an arithmetic
sequence? Surprisingly, the answer is no. To construct a counter-example,
start with 0. Then for the next term in the sequence, take the *smallest
possible integer* that doesn't cause an arithmetic sequence to form in
the sequence constructed thus far. (There must be such an integer because
there are infinitely many integers beyond the last term, and only finitely
many possible sequences that the new term could complete.) This gives the
non-arithmetic sequence 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, ...

If the terms of an arithmetic sequence are added together the result is
an **arithmetic series**, *a*_{0} + (*a*_{0} + *d*) + ... + (*a*_{0} + (*n* – 1)*d*), the
sum of which is given by:

*S*=

_{n}*n*/2 (2

*a*

_{0}+ (

*n*– 1)

*d*) =

*n*/2 (

*a*

_{0}+

*a*

_{n})

The arithmetic mean of two terms, *a _{s}* and

*a*is given by (

_{s+2}*a*+

_{s}*a*)/2 =

_{s+2}*a*.

_{s+1}