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Number Theory

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Recursive Definition

NUMTH-MB5RFJ

Recursive definitions of functions exist throughout mathematics and mathematical induction can be a way to prove results about them. Start by giving a value of the function $f$ at $n = 1$. Next, by assuming that $f$ is defined for all integers $m$ where $1 \leq m \leq n$, we have that $f(n + 1)$ is given in terms of $f(m)$ where $m \leq n$.

Let $g: \mathbb{N} \rightarrow \mathbb{N}$ be defined by $g(1) = 1$ and $g(m + 1) = mg(m)$, $m \in \mathbb{N}$. What is $g(m)$?

A

$(m - 1)!$

B

$m!$

C

$(m + 1)!$

D

$m$