How does $(k+1)!(k+2)(k+1)$ simplify to $(k+2)!(k+1)$

Question

If $$n!=n(n-1)!$$ then $$(k+1)!= (k+1)k(k-1)!$$

and $$(k+2)!$$ would be $$(k+2)(k+1)k(k-1)!$$ or $$(k+2)(k+1)!$$ but what does the extra (k+1) do to make it (k+2)!(k+1)

Answer 1

$$5! = 5\times4\times3\times2\times1$$ But then: $$4! = 4\times3\times2\times1\implies 5 !=5\times 4!$$ So by the same logic $$(k+2)!=(k+2)(k+1)!$$

Answer 2

$(k+1)!(k+2) = (k+2)!$, as explained in your question

$(k+1)!(k+2)(k+1) = (k+2)!(k+1)$ follows, by multiplying both sides by $(k+1)$.

You can always multiply both sides of a true equation by the same thing to obtain another true equation. You can also do anything else to both sides of an equation.

Answer 3

$(K+2)!= (k+1)!(k+2)$

so $(k+1)!(k+1)(k+2)= (k+2)!(k+1)$