Inductive reasoning ,Conjecture Using Inductive Reasoning
Much of the reasoning in geometry consists of three stages.
1. Look for a Pattern Look at several examples. Use diagrams and tables to help discover a pattern.
2. Make a Conjecture Use the examples to make a general conjecture .
A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary.
3. Verify the Conjecture Use logical reasoning to verify that the conjecture is true in all cases.
Looking for patterns and making conjectures is part of a process called inductive reasoning.
Example: Making a Conjecture
Complete the conjecture.
Conjecture: The sum of the first n odd positive integers is ?
Solution: List some specific examples and look for a pattern.
Examples:
First odd positive integer: 1 = 1 2
Sum of first two odd positive integers: 1 + 3 = 4 = 2 2
Sum of first three odd positive integers: 1 + 3 + 5 = 9 = 3 2
Sum of first four odd positive integers: 1 + 3 + 5 + 7 = 16 = 4 2
Conjecture: The sum of the first n odd positive integers is n 2 .
To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counterexample.
A counterexample is an example that shows a conjecture is false. |
