Applications of mathematical induction I often see mathematical induction used to verify proofs. For example the formula for the sum of all integers up to an n. Unfortunately this says nothing about how the formula was found in the first place, and if mathematical induction played a role in the finding.
I then thought about Euclid's proof of the infinite amount of prime numbers. Induction seems to play an essential role here, but again, how do you come to the idea of multiplying all known prime numbers so far, adding one, and reasoning about the primality of this number again?
So, what exactly are the applications of mathematical induction? Just for giving alternate, maybe easier, proofs for theorems you already know they are true, and you already proved them in some other way? Or do you guess the theorem and only with induction you can show that it is actually true? Are there examples where induction was used in the first place to find a new theorem?
 A: First, the product of all primes up to $k$, plus $1$ is not necessarily a prime number. Second, what is sometimes called Euclide's proof of the infinitude of primes numbers has two popular versions. One is a proof by contradiction and uses no induction. The other can be stated in the form of an inductive proof. 
Proof by induction is a proof technique. It is essential in many many proofs of common and less common, deep as well as superficial mathematical assertions. 
Mathematical induction is certainly not merely a way to give easier proofs for things you already proved in a different way. A simple example would be the proof of general associativity in a group. You must use induction (or an equivalent formulation) to prove it. Of course, it's hardly a surprising result, so you might say it is evidently true. 
Mathematicians generally don't arbitrarily search for new theorems by randomly trying various proof techniques and just see what comes out. So, I wouldn't say that induction (or any other proof technique) was ever used to find a new theorem. A mathematician would use his/her intuition and understanding of a certain area to conjecture about something. Once the conjecture is in place, and was tested a bit against some examples, one might look for a proof. That proof might involve an inductive argument. Basically, anything goes when trying to find a proof. 
Proof by induction typically works well when the assertion to be proved can be reduced to a similar problem which in some sense is smaller. I hope this answers your question. 
