With the mention about "testing", OP seems to be talking about inductive reasoning (
https://en.wikipedia.org/wiki/Inductive_reasoning), not mathematical induction which is a form of deductive reasoning.
No OP, you can't prove a theorem with inductive reasoning. It's very nice if you can verify a conjecture for a bazillion examples and people might consider it evidence that it's likely to be true and worth pursuing further, but nobody will call it a theorem until you have a deductive proof.
