A sequence is an [[Ordered Set|ordered set]] generated by discrete [[Function|function]] $f$. The common ways to denoted a sequence is: $\huge \begin{align} S &= \set{ a_{1}, a_{2}, a_{3}, \cdots } \\ S(n) &= a_{n} \\ S &= \set{a_{n}}_{n=1}^{\infty} \\ S &= \set{f_{S}(x_{n})}_{n=1}^{\infty} \end{align} $ >[!example] Example of an infinite sequence >$\huge \begin{align} >S &= \left\{ >\frac{n-1}{n+1} >\right\}_{0}^{\infty} >\\&= > >\left\{ >0, \frac{1}{3}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \cdots >\right\} > >\end{align} $ >[!example] Solving for a closed form function of a sequence >Find a formula for the given [[Sequence]]: >$\huge >S = \left\{ >34, 31, 28, 25, 22, \cdots >\right\} >$ > >$\huge \begin{align} >a_{n+1} &= a_{n} - 3 \\ >a_{0} &= 34 \\ >\\ > >a_{n+3} &= {a_{n+2} }- 3 \\ > &= {a_{n+1} - 3}- 3 \\ \\ >a_{n+3} &= > a_{n} - >\underbrace{ 3 - 3- 3}_{3 \text{ times}} \\ \\ >\\ > >a_{{n+i}} &= a_{n} + \sum^{i} -3 >\\&= a_{n} -3i \\ > >a_{0+i} &= a_{0} - 3i\\ >a_{i} &= 34 - 3i\\ >\\ > > >S(n)= >a_{n} &= \boxed{ >34 - 3n}\\ >\end{align} >$