<aside> 🔖 A sequence is a special kind of function, where the domain is a subset of $\Z$(usually $\N$ or $\Z^+$)
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In this case we can list the domain possibly with ‘…’ in both ends.
We can evaluate the function at all these listed points to get a list of evalutations.
Ex.
$$ f(0),f(1),f(2),f(3),... $$
lists all evaluations of $f:\N\rarr \mathcal{S}$(any S will do)
We often use subscript notation instead of function notation, in this case.
i.e. $a_0,a_1,a_2,…,a_n,…$ or $\{a_n\}$ if we want to talk about whole sequence.
There are too many sequences for it to be possible to write a computer program to output each.
In particular, there is a sequence that it is not even possible to describe.
On the flip side, some sequences have really short descriptions, and we can say a lot about them.
Ex.
Your 1st sequence was probably the counting numbers i.e. $a_n=n$ or $1,2,3,…,$
The even numbers, in order i.e. count by $2’$s.
The odd numbers.
An arithmetic sequence is of the form
$$ \begin{aligned}\underline{a},a+&\underline{d},a+2d,…a+nd,…\\\darr\hspace{0.3in}&\text{the common differnce}\\\text{the intial term}\end{aligned} $$
A geometric sequence is of the form is of the form
$$ \begin{aligned}\underline{a},&&a\underline{r},ar^2,ar^3,...ar^n,...\\\darr&&\text{the common ration}\\\text{the initial term} \end{aligned} $$
Ex.
$1,2,4,8,16,32,64…$
These can be combined to give arithmetic-geometric sequences
$a,(a+d)r,(a+2d)r^2,…$