Factorial solve without substitution

Question

I'm in stuck with this dimostration. I've got $$\frac{n!}{(n+1)!}$$ and it's must be $$\frac{1}{n+1}$$ If I put n=3, I've got $$\frac{3!}{(3+1)!}=\frac{1}{4}=\frac{1}{3+1}$$ and it's correct. But I can't prove the resolution without replacing, like in this case that I put n=3. Can someone explain to me how to do it?

Answer 1

Recall the definition of factorial, $n! = 1 \times 2 \times \ldots \times n$. So we have $(n+1)! = 1 \times 2 \times \ldots \times n \times (n+1) = n! \times (n+1)$. But from $n! \times (n+1) = (n+1)!$ we simply rearrange to get the desired result.