**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.

1. Look for 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

1

^{2}, 2

^{2}, 3

^{2}, 4

^{2}

Therefore we can make a conjecture that the sum of the first n odd positive integers is

*n*

^{2}.

**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.