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?

Let's prove that all objects in the universe are the same color. We'll prove by induction that given any set of $n$ objects, all of them are the same color.
Inductive step: Given a set $\{1, ... n+1\}$ of objects, we know that $\{1, ... n\}$ are all the same color $c_1$, and that $\{2, ... n+1\}$ are all the same color $c_2$. But if $n\geq 2$ these two sets overlap, so $c_1$ and $c_2$ must be the same. Therefore $\{1, ... n+1\}$ are all the same color.
Base case: If $n=1$, it's obvious. If $n=2$, then the two elements are the same color because... er...
Base case left as an exercice to the reader.
