Recursion/Induction/Remark

Closely related with induction is the principle of recursive definition. Here, one would like to define for every natural number ${}n$ a mathematical expression. This can be done by assigning to ${}0$ explicitly an expression and to describe how the expression for ${}n+1$ might be computed from the expression for ${}n$. This rule is called the recursive step. The inductive structure of the natural numbers ensures that for every natural number a unique expression is determined. For example, one can define an expression ${}F(n)$ by the initial step

${}F(0):=7\,$

and the recursive step

${}F(n+1):=F(n)\cdot n-n^{2}+3\,.$