Let [tex]a_n[/tex] denote the factorial of any non-negative integer [tex]n[/tex].
By convention, we take [tex]0!=1[/tex], so that [tex]a_0=1[/tex]. As [tex]n!=n\cdot(n-1)\cdot\cdots\cdot2\cdot1[/tex] by definition, we can obtain the next integer's factorial, [tex]a_{n+1}=(n+1)![/tex], by multiplying the previous term [tex]a_n=n![/tex] by the next integer.
The recursive definition for the factorial function can then be given by
[tex]\begin{cases}a_0=1\\a_{n+1}=(n+1)a_n&\text{for }n\ge0\end{cases}[/tex]
