Question

Provide a recursive definition of the function f(n) = (n+1)!.

Answer 1

See definition below

Explanation:

Since we have to give a recursive definition, we must give a initial value f(0). Additionally, the value of f(n) must depend on the value of f(n-1) for all n≥1.

The required value of f(0) is (0+1)!=1!=1.

Now, the factorial itself is a recursive function, because (n+1)!=(n+1)n!. In terms of f, this means that f(n)=(n+1)f(n-1) for all n≥1.

Then, our definition is: f:N→N is defined by

1. f(0)=1.
2. For n≥1, f(n)=(n+1)f(n-1).