How to Explain Induction to a friend who complete confused?

Question

How would you explain to this person why it is that induction actually does work. In my follow up responses to you, I may ask you questions that your friend might as in response to your explanation. Try to phrase those responses as though you were responding to somebody who was very confused.

Answer 1

If someone is totally confused, the explanation ought to be undertaken step by step.

 

The idea is based on an IDENTITY.  Suppose that a proposition (e.g. formula) depending on positive integer k is true.

 

IF I CAN SHOW THAT IT NECESSARILY FOLLOWS that the Proposition must also be true for k+1 (instead of just k)--this is a GENERALIZATION--then I would test the proposition for k=1.  (This would be considered the starter-value; the truth of the proposition when k = 1 is a simple empirical demonstration.)

 

If that's true, then our generalization (Prop. k true implies prop. k+1 MUST ALSO BE TRUE), then the proposition is true, consequently, for k=2.

 

It being true for k=2, our generalization is used, again, showing that it has to be true for k=3. (We keep increasing k by 1 each time that we apply the generalization implication).   And so, reasoning thus ad infinitum, the proposition is true for all positive integers.