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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, and $ a_k $ being the term, usually found with a function $ a_k = f(k) $ or rule describing them.

An example would be

$ a_k = 3^k $
$ (a_k)_{m=1}^\infty=(3,9,27,81,243,\cdots) $

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 = a_{k-1}+a_{k-2} $
$ (a_k)_{m=0}^{\infty}=(0,1,1,2,3,5,8,\cdots) $

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

$ a_k = 2k+1 $
$ (a_k)_{m=0}^{\infty}=(1,3,5,7,\cdots) $

Geometric sequences take the form

$ a_k = ar^k $
$ (a_k)_{m=0}^{\infty}=(a,ar,ar^2,ar^3,\cdots) $

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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