I have a question on Mathematical Induction.
Ok, when we are presented let's take a concrete example, with a sum of finite numbers and say we want to prove a result, e.g. that:
$ S_n = 1 + 4 + 9 + 16 +\cdots + n^2 = \frac{n(n+1)(2n+1)}{6} $
for integers $n\geq 1$
we can use mathematical induction to prove it.
But did the result to be proven came up from induction in the first place?
Or induction is just a tool to prove an outcome, fast, that we originally concluded by other means?
Note: The $S_n$ above is an example. My question is generic on induction principle.
Thank you.