When you're proving something, does it matter what you're proving T (sub n) is?

Question

When you're proving something, does it matter what you're proving T (sub n) is, as long as you have the equation you're supposed to prove? All you have to do is plug k+1 into the equation, and simplify, right?

What I mean is, you don't have to know what the equation is for to prove it's true, do you?

Answer 1

Presumably you're talking about proof by induction here...

I think most mathematicians would say that the thing that you're proving T_n to be *is* what's on the other side of the equation, so your question would make no sense. The presence or absence of any particular meaning attached to it is irrelevant.

I suspect that this is what you're actually asking - whether it's necessary to know the interpretation of what you're proving in order to prove it. The answer is: definitely not. Indeed, there's no requirement for there to be a single interpretation of it, or even any interpretation at all; it can just be a random mathematical expression as far as we are concerned.

The usefulness of this is that sometimes the same thing arises in many different contexts. We don't have to go and replicate the proof for each one; we can just say "This has the form , so we know the answer is " and not need to worry about whether we're discussing raindrops or skyscrapers.