In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
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Examples and notation
There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.
In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing. For example a sequence might be written as (a_{1}, a_{2}, a_{2}, … ), or (b_{0}, b_{1}, b_{2}, … ), or (c_{0}, c_{2}, c_{4}, … ), depending on what is useful in the application.
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