Inductive reasoning: conjectures and counterexamples

0:00Can a pattern that keeps working prove a rule?

0:04Add the odd numbers in order. 1 plus 3 is 4. 1 plus 3 plus 5 is 9. 1 plus 3 plus 5 plus 7 is 16.

0:16Each total is a square number: 2 odd numbers give 2 squared, 3 give 3 squared, and 4 give 4 squared.

0:25Inductive reasoning means using a pattern in examples to suggest a general rule.

0:31A conjecture is a statement that seems true but is not yet proved. Ours: the first n odd numbers add up to n squared.

0:42The next case works too: adding 9 gives 25, which is 5 squared.

0:48Now a warning. Look at n squared plus n plus 41.

0:54A prime number is a whole number with exactly two factors: 1 and itself.

1:00When n is 0 we get 41, then 1 gives 43, 2 gives 47 and 3 gives 53. All are prime.

1:12In fact, every n from 0 to 39 gives a prime. At 39 it gives 1601, which is prime.

1:23That is 40 primes in a row, so we might conjecture it is always prime.

1:29Now try n equals 40: 1600 plus 40 plus 41 is 1681, which is 41 times 41. Not prime.

1:43So n equals 40 is a counterexample: a case where the statement is false. One is enough to disprove it.

1:51A common mistake is treating a pattern that has worked so far as a proof. 40 examples were not enough.

2:00The odd number rule is in fact true, but only a proof can show it works for every n.

2:06Many examples can suggest a rule, but one counterexample is enough to disprove it.