I recently learned about strong induction, and had a couple questions. First, on sites such as this one: http://www.mathblog.dk/strong-induction/ , it is said that using strong induction requires more bases case. (This makes sense to me, as you are making a bigger claim) When I asked my professor, though, he said there are no more base cases needed. In this light, I don't see why anyone would ever go through the effort of even attempting 'weak' induction, as it is included in strong induction, and no extra steps of work are needed-only a larger assumption.
Strong induction is the following:
Weak induction is the following:
Usually, it is considered more elegant to assume the least amount of things possible. So if you can finish your work by using only weak induction, it is preferred. As you pointed out, you could use strong induction instead of weak induction all the time.