DWG NO. 02.1 — Lesson 1 of 5

Inductive Reasoning and Conjectures

Unit 2: Reasoning and Proof · ~15–30 min

Objective

Use inductive reasoning to spot a pattern in a set of examples, state it as a conjecture, and test the conjecture with a counterexample.

Inductive reasoning

Inductive reasoning starts with specific examples — numbers, figures, measurements — and looks for a pattern among them. Once a pattern seems to hold, you state it as a general rule, called a conjecture. Inductive reasoning is how most mathematical ideas get discovered in the first place. But spotting a pattern isn't the same as proving it always works — that's the job of deductive reasoning, which is coming up in Lesson 2.3.

Making a conjecture

To form a conjecture: gather several examples, look for what stays the same or changes in a predictable way, write the pattern as a general statement, and test it against a few more cases before you trust it.

Dot counts 1, 3, 6, … grow by one more each stage — the pattern behind the conjecture dots(n) = n(n+1)÷2.

Counterexamples

A conjecture only needs to fail once to be disproved. A single case that breaks the pattern is called a counterexample, and finding one is enough to reject the whole conjecture — no matter how many earlier cases it worked for.

Worked Example 1 · Extending a pattern

ProblemThe diagram above shows dot counts 1, 3, 6 for stages 1, 2, 3. Conjecture a formula for the number of dots at stage n, then predict stage 4.

1.Look at the differences between consecutive counts: 3−1=2, 6−3=3. Each stage adds one more dot than the last increase — consistent with adding a new, longer row each time.

2.Following that pattern, stage 4 should add a row of 4 dots to the 6 already there: 6+4=10.

3.Test a general formula: dots(n) = n(n+1)÷2. Check n=3: 3×4÷2=6 ✓. Check n=4: 4×5÷2=10 ✓, matching the prediction.

Stage 4 has 10 dots • conjecture: dots(n) = n(n+1)÷2

Worked Example 2 · Disproving with a counterexample

ProblemTest the conjecture: “The sum of two prime numbers is always even.”

1.Try a few cases: 3+5=8 (even), 5+7=12 (even), 3+7=10 (even) — the pattern seems to hold so far.

2.But 2 is also prime, and it's the only even prime. Try 2+3=5, which is odd.

3.That single case breaks the pattern, so it's a valid counterexample — the conjecture is false as stated.

Counterexample: 2+3=5 (odd) • conjecture is false

Guided practice

1.Find the next term and conjecture a general rule for the sequence 2, 5, 8, 11, …
Show answer
Next term: 14 (each term adds 3). General rule: \(a_{n}\) = 3n − 1, since \(a_{1}\)=3(1)−1=2, \(a_{2}\)=3(2)−1=5, and so on.
2.A 1×1 grid has 1 unit square, a 2×2 grid has 4 unit squares, and a 3×3 grid has 9 unit squares. Conjecture a rule for the number of unit squares in an n×n grid.
Show answer
squares(n) = n². The counts 1, 4, 9 are exactly 1², 2², 3².
3.Find a counterexample to disprove: “All quadrilaterals have four congruent sides.”
Show answer
Any non-square rectangle works, e.g. a 4-by-2 rectangle — its sides are not all congruent, but it's still a quadrilateral.
4.Test the conjecture “The sum of two odd numbers is always odd” and state whether it's true or false.
Show answer
False. Counterexample: 3+5=8, which is even, not odd.
5.Can inductive reasoning ever prove a conjecture is true for every possible case? Explain.
Show answer
No. Inductive reasoning only shows a pattern held for the cases you checked — there could always be an untested case that breaks it. Only a deductive proof, built from accepted facts, can guarantee a statement is true in every case.

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