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A sequence is an ordered set of objects. A sequence that goes on forever is called an infinite sequence, whereas one that does not is called a finite sequence. The sum of a sequence is called a series.

Sequences are usually denoted as

\{a_k\}_m^n

with k being the term number and m,n being the bounds of the series.

An example would be

\{a_k\}_{m=1}^\infty=3^k=3,9,27,81,243,\ldots

Sequences described with the previous terms are called recursive sequences. For instance, the Fibonacci numbers can be described as

a_0=0,a_1=1
\{a_k\}_{m=0}^\infty=a_{k-1}+a_{k-2}

Two common types are arithmetic and geometric sequences. Arithmetic sequences have a given difference between each term. For example,

\{a_k\}_{m=0}^\infty=2k+1=1,3,5,7,\ldots

Geometric sequences take the form

\{a_k\}_{m=0}^\infty=ar^k

Where r is a common ratio.

Sets vs Sequences

Unlike a set , sequences allow repeats and the order matters.

Formal definition

Let S be a set
Let \N be the set of natural numbers.
Then, a mapping a:\N\to S is called a sequence of elements of S . The image of an element of under a (that is a(i)) is denoted as S_i , where i\in\N .

An equivalent definition is an indexed family indexed by the natural numbers.

While the formal definition of a sequence is a treats a sequence as a function, in practice they are treated somewhat like a set with "order". 

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