Mathematical Induction

When we prove by mathematical induction, after assuming S(k) is true, can we prove that S(k-1) or S(k+2) etc. is true instead of S(k+1) to conclude that the statement S(n) is true for all natural number n?

You can't use S(k+2), as you have assumed S(k) is true, if you prove S(k+2) is true, that means S(k+1) may not be true..., you will skip some terms...

Actually, we have proved S(1) is true first, then we assume S(k) is true , so that we prove s(k+1) is true, for k=1 , it is true, k=1+1 is also true , so that we can further deduce all the natural nos.

But for S(k-1), if k=1 is true, and assume S(k) is true, we find out that S(k-1) is true, then when k=1, we can only find out that S(0) is true, and we can further deduce that S(-1), S(-2)...... and so on, it's true, so in my opinion, i won't use this approach

In some cases, you can use this, while you can prove that S(-k) is also true...

For S(k+2), if I prove that it's true for S(1) and S(2), is it true that S(n) is true for all positive n?

u can say so, but we seldom use this, as if we can prove S(k+2), we can easily prove S(k+1)