As you already know, mathematical induction is a mathematical proof technique that is used to prove that some property ##P(n)## holds true for every natural number. I think that the key here is to understand what is induction exactly and do as many examples / exercises as you can. This will vastly help you gain the required momentum in order to recognize where is induction needed and apply the concept correctly. There is a multitude of examples you can find using a simple google search. Also, you can take a look at this cmu.edu page for some problems beyond the usual (very basic) ones - it is Computer Science notes so it has some relevant to CS examples.Yes, on how to recognize it and why it works. (a farmer which has ##n+1## sheeps of which ##n##are black doesn't imply all are, I probably have a very shallow understanding of it)
Mathematical induction is a very important method regarding proofs but (obviously) it is not a litmus test. For this particular question, I would hint to think of the problem in terms of functions i.e. using mostly Calculus.To put it sarcastically, "I'm a simple man, I see ##n## I use induction".