I have a question regarding strong induction. I've seen examples on proofs that assume that P(n) is true for all n that is smaller or equal than k and thereby dealing with k+1 in the inductive step followed. In some other other case i have seen examples where it is assumed that P(n) is true for all n < k followed by induction proof on the k.
Can someone tell me why this is different and which one is correct approach?
It varies depending on the problem.
Ordinary induction proves n assuming the truth for n-1.
Mildly strong induction, to prove truth for n, might have to assume the truth for n-1 and n-2.
Somewhat strong induction, to prove truth for n, might have to assume the truth for a fixed number of predecessors n-1, n-2, ..., n-k.
Really strong induction, to prove truth for n, might have to assume the truth for a number of predecessors that depends on n: n-1, n-2, ..., n-k(n).
And finally, for the traditionally strong induction, to prove truth for n, might have to assume the truth for all of its predecessors: n-1, n-2, ..., 2, 1.
It all depends on what you are trying to prove.