once that is shown, it would mean that for any value of "n" that is picked, the next one is true. since it is true for some beginning case (usually "n"=1), it's true for the next one ("n"=2). and since it is true for 2, it must be true for 3. and since it is true for 3, it must be true for 4, etc. induction shows that it is always true, precisely because it is true for whatever comes after any given number.