Induction hypothesis for $n-1$

Question

I have a question about mathematical induction. Let $A(n)$ be a statement dependent of a natural number $n$. The task consist to show that the statement is true for all $n\in \mathbb{N}$ by mathematical induction. On the induction hypothesis I assume that the statement is true for a particular $n\in \mathbb{N}$, then I have to show on the induction step that with this assumption the statement is also true for $n+1$.

So my question is, can I use on the induction step that also the statement is true for $n-1$? or maybe also for $n-k$ for $k<n$?

For example, if I need to show a statement of a recurrence formula that contains $n$ and $n-1$ on the recurrence for $n+1$.

Thanks for yours answers.

Answer 1

What you want to do here is to use strong mathematical induction. It turns out that the principle of mathematical induction and the principle of strong mathematical induction are equivalent, so yes you're allowed to do that (as long as you accept mathematical induction).