How many examples do I need before I can trust a conjecture?

No number of examples is enough for certainty. $n^2 + n + 41$ works $40$ times and then fails. More examples, and varied ones, make a conjecture more believable, but only a proof makes it certain.

Is mathematical induction a kind of inductive reasoning?

No. Despite the name, mathematical induction is a deductive proof: you prove a base case and prove that each case implies the next, so the statement is certain for all whole numbers.

Are there famous conjectures nobody has proven yet?

Yes. Goldbach's conjecture says every even number greater than $2$ is the sum of two primes, like $10 = 3 + 7$. Computers have checked it for every even number up to $4 \times 10^{18}$, but it is still unproven.

Teacher Resources

Inductive Reasoning

Learn to make a conjecture from specific cases, test it with new ones, and break it with a counterexample.

Intermediate25 minLesson

Printable worksheet

Definition

Inductive reasoning means looking at specific cases, noticing a pattern and making a general guess about it. That guess is called a conjecture.

Inductive reasoning works in four steps:

1.Observe specific cases: $1 = 1$ , $1 + 3 = 4$ , $1 + 3 + 5 = 9$
2.Make a conjecture: the sum of the first $n$ odd numbers is $n^2$
3.Test it with new cases: $1 + 3 + 5 + 7 = 16 = 4^2$ , still true
4.Prove it or break it: a proof shows it is always true, one counterexample shows it is false

A conjecture reached by inductive reasoning is only probably true, never certain. The expression

n^2 + n + 41

gives a prime number for

n = 0, 1, 2, \ldots, 39

, forty cases in a row, yet for

n = 40

it gives

1681 = 41 \times 41

, which is not prime.

Deductive reasoning goes the other way: it starts from facts that are known to be true and reaches a conclusion that must be true. Inductive reasoning finds ideas, deductive reasoning proves them.

Careful with the names: mathematical induction (from the Proofs topic) is a deductive proof method, not inductive reasoning.

Try it now

Which of these is an example of inductive reasoning?

Worked Examples

Add the first few odd numbers: $1$ , $1 + 3$ , $1 + 3 + 5$ , $1 + 3 + 5 + 7$ . Make a conjecture and use it to predict the sum of the first $10$ odd numbers.

Observe the cases

$1 = 1$ , $1 + 3 = 4$ , $1 + 3 + 5 = 9$ , $1 + 3 + 5 + 7 = 16$ → Sums: $1, 4, 9, 16$

Look for a pattern

$1 = 1^2$ , $4 = 2^2$ , $9 = 3^2$ , $16 = 4^2$ → Every sum is a square number

Make a conjecture

With $n$ odd numbers the sum seems to be $n \times n$ → Sum of the first $n$ odd numbers $= n^2$

Test it with a new case

$1 + 3 + 5 + 7 + 9 = 25 = 5^2$ → The conjecture survives the test

Use it to predict

$n = 10$ : $10^2 = 100$

Answer: Conjecture: the sum of the first $n$ odd numbers is $n^2$ . The sum of the first $10$ odd numbers should be $100$ (and this conjecture has in fact been proven).

Common Mistakes

Thinking a conjecture is proven because it worked in many cases

Why it's wrong: Examples can only support a conjecture. $n^2 + n + 41$ is prime for $40$ values in a row and still fails at $n = 40$ .

Correct: Call it a conjecture until it is proven. Say "probably true", not "true".

Believing one counterexample is not enough to reject a conjecture

Why it's wrong: A conjecture claims something about every case. If even one case fails, the claim "always" is false.

Correct: One counterexample is enough. $2$ is prime and even, so "all primes are odd" is false.

Testing only easy, positive whole numbers

Why it's wrong: Many conjectures break at $0$ , at negative numbers or at fractions. "Squaring makes a number bigger" works for $2, 3, 10$ but fails for $\frac{1}{2}$ : $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ .

Correct: Test special cases on purpose: $0$ , $1$ , negatives, fractions and large numbers.

Confusing mathematical induction with inductive reasoning

Why it's wrong: The names sound alike, but mathematical induction is a deductive proof method with a base case and an induction step.

Correct: Inductive reasoning makes a conjecture from examples. Mathematical induction proves a statement for all whole numbers.

Interactive Visual

Sequence Explorer

First Term (a₁ = 1)

Common Difference (d = 2)

11,

13,

, ...

Formula

a_n = 1 + 2(n - 1) = 2n - 1

Sum Formula: S_8 = 8/2 × (2×1 + 7×2) = 64

Partial sums

S₁ = 1S₂ = 4S₃ = 9S₄ = 16S₅ = 25S₆ = 36S₇ = 49S₈ = 64

a_1

a_5

a_10

a_20

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems

Problem 1 of 18

Easy

Which of these is an example of inductive reasoning?

Practice Problems: Inductive Reasoning

1. Which of these is an example of inductive reasoning?
2. Use the conjecture "the sum of the first n odd numbers is n²" to find 1 + 3 + 5 + 7 + 9 + 11 without adding term by term.
3. Dots are arranged in triangles: figure 1 has 1 dot, figure 2 has 3 dots, figure 3 has 6 dots and figure 4 has 10 dots. If the pattern continues, how many dots does figure 6 have?
4. Which number is a counterexample to the conjecture "all prime numbers are odd"?
5. Conjecture: "doubling a number always makes it bigger." Which number is equal to its own double and is therefore a counterexample?
6. A sunflower is 2 cm tall after week 1, 5 cm after week 2, 8 cm after week 3 and 11 cm after week 4. If the pattern continues, how tall will it be after week 8 (in cm)?
7. Conjecture: the sum of the first n odd numbers is n². Test it with n = 5, then use it to predict the sum of the first 20 odd numbers.
8. Tom adds 1 + 5, 9 + 3 and 15 + 7, gets an even number each time and concludes: "the sum of two odd numbers is always even." Which kind of reasoning did Tom use?
9. A triangle has 0 diagonals, a quadrilateral 2, a pentagon 5, a hexagon 9, a heptagon 14 and an octagon 20. Continue the pattern: how many diagonals does a nonagon (9 sides) have?
10. Look at these products: 11 × 11 = 121, 111 × 111 = 12321, 1111 × 1111 = 1234321. Make a conjecture and predict 11111 × 11111.
11. Lena claims: "squaring a number always makes it bigger." Test her claim with 1/2: what is (1/2)²? Write your answer as a fraction.
12. The expression n² + n + 41 gives a prime number for every n from 0 to 39. For n = 40 it gives 1681. Which statement is correct?
13. At a party everyone shakes hands with everyone else exactly once. With 2 people there is 1 handshake, with 3 people 3 handshakes and with 4 people 6 handshakes. How many handshakes are there with 10 people?
14. Test the conjecture "n² + n + 41 is always prime" step by step and find the value that breaks it at n = 40.
15. From the cases 0, 2, 5, 9 (polygons with 3 to 6 sides) we made the conjecture that a polygon with n sides has (n(n - 3))/2 diagonals. Use it: how many diagonals does a polygon with 12 sides have?
16. Sam notices: 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 2^7 - 1 = 127, all prime. He conjectures: "if p is prime, then 2^p - 1 is prime." Which value of p is a counterexample?
17. Put points on a circle and join every pair of points with a straight line. Count the regions inside the circle: 1 point gives 1 region, 2 points give 2, 3 points give 4, 4 points give 8, 5 points give 16. A careful drawing with 6 points, placed so that no three lines meet at one point, shows 31 regions. Test the doubling conjecture: by how many regions does its prediction miss?
18. Conjecture: "the sum of three consecutive whole numbers is always 3 times the middle number." Test it inductively, then use the deductive version to find 49 + 50 + 51.

Why It Matters

Inductive reasoning is how most new ideas begin, in math and in life:

Weather: forecasters look at many past days with similar clouds, wind and pressure and predict what will probably happen tomorrow
Science: a scientist repeats an experiment many times and states a law that fits every result, until a new result breaks it
Mathematics: famous theorems started as conjectures found by checking examples; some, like Goldbach's conjecture, are still unproven
Everyday life: if the bus has been late every morning this week, you leave home earlier tomorrow

Knowing its limits matters just as much: one counterexample is enough to sink a conjecture, no matter how many examples supported it.

Real World Applications

Weather Forecasts

Forecasters compare today with many past days that looked similar. What usually happened next becomes the forecast, which is why forecasts give chances, not certainties.

Example:

If on $80$ of $100$ past days with these clouds and this wind it rained the next day, the forecast says an $80\%$ chance of rain.

1Try It Yourself

The morning temperature was $8°C$ on Monday, $10°C$ on Tuesday, $12°C$ on Wednesday and $14°C$ on Thursday.

If the pattern goes on, what will the morning temperature be on Saturday?

Step 1: Write the mathematical expression

Start at Thursday's $14$ and add $2$ days of $2$ degrees:

Experiments in Science

Scientists repeat measurements, look for a pattern and state a rule. The rule is trusted as long as no experiment breaks it.

Example:

Hanging $1$ , $2$ and $3$ equal weights on a spring stretches it $3$ cm, $6$ cm and $9$ cm. The conjecture: the stretch is $3$ cm per weight. With too many weights the spring is damaged and the rule fails.

2Try It Yourself

A spring stretches $3$ cm with $1$ weight, $6$ cm with $2$ weights and $9$ cm with $3$ weights.

Using the conjecture, how far will it stretch with $7$ weights?

Step 1: Write the mathematical expression

Multiply the stretch per weight by the number of weights:

Planning a Tournament

In a round robin tournament every team plays every other team once. Organizers can find the number of games from small cases and a conjecture.

Example:

$2$ teams play $1$ game, $3$ teams play $3$ games, $4$ teams play $6$ games. The conjecture $\frac{n(n - 1)}{2}$ fits all three cases, and here it can even be proven: each of $n$ teams plays $n - 1$ others, and every game is counted twice.

3Try It Yourself

A school chess club has $8$ players. Each player plays every other player exactly once.

How many games will be played?

Step 1: Write the mathematical expression

Use the conjecture $\frac{n(n - 1)}{2}$ with $n = 8$ :

Key Takeaways

1Inductive reasoning goes from specific cases to a general conjecture
2Test a conjecture with new cases, especially $0$ , negatives, fractions and large numbers
3Many examples make a conjecture probably true, but they never prove it
4One counterexample is enough to show a conjecture is false
5Deductive reasoning proves a conjecture: inductive reasoning finds the idea, deductive reasoning makes it certain
6Mathematical induction is a deductive proof method, not inductive reasoning

Frequently Asked Questions

Because it is how we find ideas worth proving. Deductive reasoning needs a statement to start from, and that statement usually comes from noticing a pattern in examples.

Glossary

Inductive reasoning: Reasoning from specific cases to a general conclusion that is probably, but not certainly, true
Conjecture: A general statement that seems true from examples but has not been proven
Counterexample: One case for which a statement is false; it shows the statement is not always true
Deductive reasoning: Reasoning from known facts and rules to a conclusion that must be true
Proof: A deductive argument that shows a statement is true in every case

Formula Card

The inductive cycle

\text{observe} \to \text{conjecture} \to \text{test} \to \text{prove or disprove}

Examples lead to a guess; testing and proof decide whether it holds

Sum of the first n odd numbers

1 + 3 + 5 + \cdots + (2n - 1) = n^2

Found by inductive reasoning from examples, later proven

Diagonals of a polygon

d = \frac{n(n - 3)}{2}

Number of diagonals of a polygon with $n$ sides

Inductive Reasoning

Definition

Worked Examples

Common Mistakes

Interactive Visual

Sequence Explorer

Formula

Partial sums

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Weather Forecasts

Experiments in Science

Planning a Tournament

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

The inductive cycle

Sum of the first n odd numbers

Diagonals of a polygon

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Converse and Inverse