What is the purpose of the first test in an inductive proof?

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.

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).

The induction proof is, in a metaphor, a game of dominos. Your goal is to make the pieces fall over, where if the $n$th domino falls, so will the $(n+1)$st. However, if you don't knock down the beginning domino, none may fall over.

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An example of what goes wrong without appropriate base case...

Claim: For all $n \in \Bbb{N}=\{1, 2, 3, ...\}$, $n \geq 1000$.

Proof without base case: If $n \geq 1000$ the $n+1 > n \geq 1000$.

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An example of what goes totally wrong...

Claim: For all $n \in \Bbb{N}$, $n > n$.

Proof without base case: If $n > n$, then by adding $1$ to both sides we see $n+1>n+1$. Therefore, by baseless induction, we've "proved" the claim.