A sequence is a [[Function]] $S(n) : \Z^+\to S$, where $S$ is a [[Set]]. 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]- A Sequence can be a list of numbers in a specific order. >[!example]- >$\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} $ ### Solving for a closed form [[Function]] of a [[Sequence]] >[!example] 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} $