Chapter 1.1 - Patterns and Inductive Reasoning
A. Using Inductive Reasoning
1. Look for a pattern. - look at several examples. Use diagrams and tables to help discover a pattern.
2. Make a Conjecture. - a conjecture is an unproven statement that is based on observation.
3. Verify the Conjecture - use logical reasoning to verify that the conjecture is true in all cases.
Example - the sum of the first n odd positive integers is...
first odd positive integer is 1 = 1
sum of first two odd positive integers is 1 + 3 = 4
sum of first three odd positive integers is 1 + 3 + 5 = 9
sum of first four odd positive integers is 1 + 3 +5 + 7 = 16
looking at these answers: 1, 4, 9, 16 we see they are perfect squares so
12, 22, 32, 42
Therefore we can make a conjecture that the sum of the first n odd positive integers is n2 .
B. Counterexample - is an example that shows a conjecture is false.
Example: All prime numbers are odd.
Since 2 is a prime number but 2 is even this would be a counterexample.