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?

Imagine a pond with an infinite linear progression of lily pads. You have a frog who, if he hops on one pad, he is guaranteed to hop on the next one.  If he hops on the first pad, he'll visit them all. But if he never makes the first lilypad, all bets are off.
