The strong form of the principle of mathematical induction and the ordinary form.

[kntsy]

Can i use the strong form every time? Why do people still use the ordinary form?

[Gib Z]

People as in the average math student, or mathematicians?

The average math student wouldn't use it every time because often the weak form is all they need and it ends up being simpler or quicker.

A Mathematician never likes to use a stronger result than he/she has to. They like to use only the most basic ideas required to get the result.

Of course, you MAY use it all the time if you wish.

[Landau]

Actually, despite the name, the "strong" and "ordinary" forms of induction are (logically) equivalent. So certainly you can use either depending on the problem or you own taste.