Name: ______________________ Date: ________________ Foundations of Mathematics 11 Chapter 1- Inductive and Deductive Reasoning 1.3 Using Reasoning to Find a Counterexample Today’s Goal: To find examples that contradict given conjectures. Let’s define Counterexample: In order to disprove or make a conjecture false we must find one example of the statement that does not satisfy the conjecture. Conjecture: Natural numbers can be written as the sum of consecutive numbers. Here are some examples: 9 = 4 + 5 6 = 1 + 2 + 3 12 = 3 + 4 + 5 10 = 1 + 2 + 3 + 4 22 = 4 + 5 + 6 + 7 29 = 14 + 15 Find a counterexample to this conjecture. How can you be sure that your counterexample is correct? Example: Matt found a interesting numeric pattern: 1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 Matt thinks that this pattern will continue. Search for a counterexample to Matt’s conjecture. A B 1 1 x 8 + 1 9 2 12 x 8 + 2 98 3 123 x 8 + 3 987 4 1234 x 8 + 4 9876 5 6 7 8 9 10 The pattern holds true until the ______________digit is added we have a counterexample
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