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arithmetic sequence
Also known as an arithmetic progression, 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 an = dn + a0 (n = 1, 2, 3,...).
Does every increasing sequence of integers have to contain an arithmetic progression? 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 progression 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 progressions 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:
Related category
• MATHEMATICS
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