Three specimen worked examples for induction: Statement‑first, Base‑case‑first, Forward‑chaining

Which worked-example format should you teach or learn for a first exposure to proof by induction, and how do you construct a single short example in that format? Below are three compact specimen rows using the same basic algebraic claim so you can pick one structure, adapt it, and deliver a focused single-class example.

Style 01 — Statement‑first (museum caption)

Start by declaring the claim, then show its structure; base case follows the claim to reveal the skeleton.

Claim P(n): for all n ≥ 1, 1 + 3 + … + (2n−1) = n²

Sketch:
1) State P(n).
2) Verify base n=1: 1 = 1².
3) Assume P(k): 1+…+(2k−1)=k².
4) Show P(k+1): add 2(k+1)−1 to both sides.
   k² + 2k + 1 = (k+1)². QED.

student wonder; instructor move; learner cue

What is the general claim Print the formula; name it P(n) Sees the full skeleton

Why check n=1? Show direct substitution Confirms where induction anchors

Explicitly name the induction hypothesis

Teaching notes: Use when you want the formal skeleton visible immediately; good for showing the canonical order of steps. Keep algebra short and call out where the induction hypothesis is applied.

Takeaway: Statement‑first exposes the full proof shape quickly; students see what each step must accomplish.

Style 02 — Base‑case‑first (museum caption)

Begin with an explicit base verification, then generalize to the statement; lowers initial symbolic load.

Base: n=1 → 1 = 1² (checked)

Quick computation for n=2:
1 + 3 = 4 = 2²

From these concrete verifications, propose:
Claim P(n): 1 + … + (2n−1) = n² for n ≥ 1.

student question; instructor action; learner cue

Does it hold for small n? Compute n=1 and n=2 aloud Builds concrete trust

What general pattern fits? Introduce the general formula Student maps examples to symbol

Emphasize why base lowers anxiety

Teaching notes: Use when students are unfamiliar with algebraic shorthand or when initial cognitive load is a concern. Follow by a concise induction step after naming P(n).

Takeaway: Base‑case‑first grounds the claim in calculation; it trades early formalism for immediate intuition.

Style 03 — Forward‑chaining (museum caption)

Lead with a chain of concrete n values that suggests the conjecture; then formalize the hypothesis and perform the inductive step.

Compute:
n=1 → 1 = 1
n=2 → 1+3 = 4
n=3 → 1+3+5 = 9

Conjecture: P(n): sum of first n odd = n²

Formalize:
Assume P(k); then sum to k+1 = k² + 2(k+1)−1 = (k+1)². Hence P(k+1).

student prompt; instructor sequence; learner cue

What pattern emerges? Show n=1,2,3 quickly Student conjectures n²

How to prove generally? State induction hypothesis and add next term Sees transfer from k to k+1

Highlight transfer cue: "add the next odd term"

Teaching notes: Use when visual pattern recognition is helpful. Warn students against treating a few cases as proof; explicitly move from examples to a formal hypothesis and then to the algebraic step.

Takeaway: Forward‑chaining builds conjecture via examples, then secures it with the induction step.

Key distinctions — quick chooser

Use these concise distinctions to choose or adapt a style for a single class session:

Transparency vs cognitive load; statement‑first exposes the whole flow; base‑case‑first reduces early symbolic burden;
Conjecture recruitment; forward‑chaining recruits pattern recognition before formal language;
Where to name the hypothesis; always label "Assume P(k)" explicitly — missing the label is a frequent student error;
Base clarity; show which n the claim quantifies over (n ≥ 1 vs n ≥ 0) to avoid domain confusion;
Transfer cueing; highlight the concrete operation that transforms k to k+1 (add one term, multiply, shift index);
When time is short; prefer statement‑first with a short algebraic step, or base‑case‑first if students need arithmetic reassurance;
Common mistakes to preempt; misreading the quantifier; treating examples as proof; failing to use the induction hypothesis correctly.