Inductive step assumption for all numbers up to $n$

Question

I know that the inductive step should be "for all $n$ (if $P(n)$ then $P(n+1)$)" and NOT "if (for all $n$ $(P(n)$)) then (for all $n$ ($P(n+1)$))" - see this answer.

But can it be like "if (for all $i$, where $i = 1, \ldots, n$, ($P(i)$)) then $P(n+1)$)"?

Answer 1

Yes, it can. It is called strong induction.

Thanks to Raj, I found answer to my question here: difference between simple induction and strong induction.