What is the purpose of the first test in an inductive proof? Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$"
Like this:


Prove that $1+2+3+...+n = \frac{n(n + 1)}{2}$ for all $n \ge 1$.

Test it for $n = 1$:
$$n = \frac{n(n + 1)}{2} \implies 1 = \frac{2}{2}\textrm{, which holds.}$$
* The rest of the proof goes here *

So, I do it all the time (like a standard). But I never really thought about why. Why do I do such test? I can see that if the test does not hold, it cannot be proven by induction, I guess. But is there another reason we do this?

We have to test the "base case" because otherwise we could "prove" things by induction that aren't true.  For instance: let's "prove" that the sum of the first $n$ even numbers is odd.  Certainly if $S = 2+4+\cdots+2n$ is odd, then $2+4+\cdots+2n+(2n+2) = S + 2n+2$ is also odd, since an even number plus an odd number is odd.
So by "induction", the sum of the first $n$ even numbers is odd.
Of course this is false, and the reason we got something false is that we didn't verify that the base case holds (it doesn't).
