Skipping the Base Case Because It's 'Obvious' — Then Discovering the Statement Was Actually False

The temptation is real. You look at the statement you are trying to prove by induction, and the base case looks so trivial you can see at a glance that it must be true. The inductive step, meanwhile, will take a page of algebra to write out cleanly. So you skip the base case — "obviously P(1) holds" — and focus your attention where the real work is.

This habit is one of the most reliably dangerous shortcuts in induction. It does not just cost neatness marks in an exam. It occasionally costs you the entire proof — because the base case was actually false, the statement itself was wrong, and you have just produced a valid-looking "proof" of a false theorem.

The documented pattern

The failure mode runs like this:

You have a statement P(n) that looks natural and clean.
You write an inductive step that is valid — the implication P(k) \Rightarrow P(k+1) genuinely holds.
You wave away the base case with a phrase like "clearly P(1) holds" or "the base case is trivial."
The actual P(1) is false. The chain of implications never fires, because it had no starting link.

When step 4 goes unnoticed, you conclude P(n) for all n — and since the inductive step is valid, any person checking your proof by scanning the algebra will miss the problem. The only check that catches it is the one you skipped.

A classic example: a false "formula"

Suppose a student guesses the formula

1 + 2 + 3 + \dots + n = \frac{n^2 + n + 2}{2}

and tries to prove it by induction. Setting aside whether the formula is right, look at the inductive step.

Assume 1 + 2 + \dots + k = \frac{k^2 + k + 2}{2}.
Then 1 + 2 + \dots + k + (k+1) = \frac{k^2 + k + 2}{2} + (k+1) = \frac{k^2 + k + 2 + 2k + 2}{2} = \frac{k^2 + 3k + 4}{2}.
We want this to equal \frac{(k+1)^2 + (k+1) + 2}{2} = \frac{k^2 + 2k + 1 + k + 1 + 2}{2} = \frac{k^2 + 3k + 4}{2}. ✓

The inductive step works out perfectly. If you stop here — "the inductive step goes through, and the base case is obvious" — you walk away believing the formula. But check n = 1:

\text{LHS} = 1, \quad \text{RHS} = \frac{1^2 + 1 + 2}{2} = \frac{4}{2} = 2.

1 \neq 2. The base case is false. The formula is wrong (the true formula is \frac{n(n+1)}{2} = \frac{n^2 + n}{2}, without the extra +2). The inductive step is valid but useless — it correctly propagates a false fact forward, saying "if the formula held at k, it would hold at k+1," without any initial truth to propagate.

This is the exact pattern of the documented error. The inductive step is airtight. The base case — which the student assumed was obvious and did not verify — silently hides the falsity of the entire claim.

Why the inductive step cannot save you: induction's logical engine is P(1) \wedge (\forall k: P(k) \Rightarrow P(k+1)) \Rightarrow \forall n: P(n). If P(1) is false, the premise of the big conditional is false, and the conclusion \forall n: P(n) is unsupported. A "proof" with a valid inductive step and a false base case proves nothing at all — yet it looks almost indistinguishable from a correct proof on the page, because the inductive-step algebra is the flashy part.

Why students skip

Three honest reasons students skip the base case:

It looks trivial. For most textbook statements, P(1) really is a one-line check. It feels like the kind of step you might as well skip to save time.
The inductive step looks harder. Attention drifts toward the step, where the actual algebra lives, and the base case feels administrative.
It is rarely emphasised in explanations. Presentations focus on the idea of the inductive step — the clever chaining — and treat the base case as a detail.

Each of these is understandable. None of them are valid reasons. The base case is where the entire chain gets its start, and therefore where the biggest possible error can hide: the error of proving an implication rule for a statement that is simply not true.

A good discipline

Three habits to build:

Always write the base case explicitly, even if it takes one line. "Check: P(1) says 1 = 1, which holds." That one line is the difference between a proof that produces a true theorem and a proof that produces nothing.
Do the base case first, before the inductive step. If the base case fails, you save yourself the work of the inductive step and — more importantly — you discover that your statement is wrong before you waste an hour defending it.
Plug in n = 1, 2, 3 numerically and confirm by arithmetic. Do not trust symbolic cleanliness. If the formula predicts \frac{n^2 + n + 2}{2}, compute this at n = 1, 2, 3 and verify it actually equals the left-hand side. This is a sanity check that takes ten seconds and catches most base-case errors.

In competition mathematics, professional mathematicians frequently test conjectures at small n before attempting to prove them. They know from experience that a surprising number of plausible formulas fail at n = 1 or n = 2 and would waste proof effort otherwise. The same habit belongs in your homework.

A subtler variant: the wrong base case

Sometimes the statement is true for all n \geq 2 but false for n = 1. A typical example:

2^n > n^2 \quad \text{for all } n \geq 1.

The inductive step works when k is large enough. But check the claimed base case n = 1: 2^1 = 2 > 1 = 1^2. Then n = 2: 2^2 = 4 > 4 = 2^2 — false, the inequality is not strict. And n = 3: 8 > 9? No. And n = 4: 16 > 16? No. The inequality actually holds strictly only for n \geq 5, where 2^5 = 32 > 25 = 5^2, and then it holds forever after.

If you set the base case as n = 1 without checking, you miss that the statement is genuinely false for n = 2, 3, 4. The fix is not to skip the base case harder; the fix is to set the base case at the smallest n for which the statement is actually true — and to verify it.

The pattern to remember

A documented, repeated error: "The inductive step works. The base case is obvious. Therefore the theorem is proved." The middle clause is the trap. What "obvious" often means in practice is "I did not check." The moment you treat the base case as administrative, you invite the possibility that the thing you are proving is not true — and you will get through the rest of the proof without noticing.

The fix is as simple as it is boring: always write P(1) down and verify it numerically, before you start the inductive step. Ten seconds of effort. It is the only step in induction that confirms the theorem is about something real.

The one-line takeaway

The base case is the only part of induction that anchors the proof to reality. Skipping it — because "it's obvious" or "the inductive step is where the action is" — occasionally produces a valid-looking proof of a false theorem. The habit of always writing and checking P(1) is cheap, boring, and the single most effective safeguard against this documented failure.