# Sequence

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A sequence, in mathematics, is an enumerated list; the elements of this list are usually referred as to the terms. Sequences may be finite or infinite.

Formally, given any set X, an infinite sequence is a function (f, say) defined on the natural numbers ${\displaystyle \{1,2,3,...\}}$, with values in X. Similarly, a finite sequence is a function f defined on ${\displaystyle \{1,2,3,...,n\}}$ with values in X. (We say that n is the length of the sequence).

In a natural way, the sequences are often represented as lists:

${\displaystyle a_{1},\,a_{2},\,a_{3},\dots }$

where, formally, ${\displaystyle a_{1}=f(1)}$, ${\displaystyle a_{2}=f(2)}$ etc. Such a list is then denoted as ${\displaystyle (a_{n})}$, with the parentheses indicating the difference between the actual sequence and a single term ${\displaystyle a_{n}}$.

Some simple examples of sequences of the natural, real, or complex numbers include (respectively)

10, 13, 10, 17,....
1.02, 1.04, 1.06,...
${\displaystyle 1+i,2+3i,3+5i,\dots }$

Often, sequences are defined by a general formula for ${\displaystyle a_{n}}$. For example, the sequence of odd naturals can be given as

${\displaystyle a_{n}=2n+1,\quad n=0,1,2,\dots }$

There is an important difference between the finite sequences and the [[set]s. For sequences, by definition, the order is significant. For example the following two sequences

1, 2, 3, 4, 5 and 5, 4, 1, 2, 3

are different, while the sets of their terms are identical:

{1, 2, 3, 4, 5} = {5, 4, 1, 2, 3}.

Moreover, due to indexing by natural numbers, a sequence can list the same term more than once. For example, the sequences

1, 2, 3, 3, 4, 4 and 1, 2, 3, 4

are different, while for the sets we have

{1, 2, 3, 3, 4, 4} = {1, 2, 3, 4}.