8064

210

$$-$$546

5040

Let $$P(n)$$ be a statement such that $n$ be any integer and $$P(1)$$ is true. Also, $$P(m)$$ is true for $$m$$, any natural number, then $$P(n)$$ is true for all integers $$n$$

Let $$P(n)$$ be a statement involving natural number $$n$$ such that $$P(1)$$ is true and $$P(m)$$ is true whenever $$P(n)$$ is true for every $$n \geq m$$, then $$P(n)$$ is true for all $$n \in N$$ (set of natural numbers)

Let $$P(n)$$ be a statement where $$n \in N$$ such that $$P(1)$$ is true and $$P(n), P(n+1)$$ also holds, then $$P(n)$$ is true $$\forall n \in N$$

Let $$P(n)$$ be a statement involving the natural number $$n$$, such that $$P(1)$$ is true and $$P(m+1)$$ is true for all $$n \leq m$$. Then, $$P(n)$$ is true for all $$n \in N$$