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, ..., 4n + 1, ... In general, the terms of an arithmetic sequence with the first term a₀ and common difference d, have the form a_n = dn + a₀ (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, a₀ + (a₀ + d) + ... + (a₀ + (n – 1)d), the sum of which is given by:

 

     S_n = n/2 (2a₀ + (n – 1)d) = n/2 (a₀ + a_n)

 

The arithmetic mean of two terms, a_s and a_s+2 is given by (a_s + a_s+2)/2 = a_s+1.