2.1.1 Using Inductive Reasoning to Make Conjectures

Key Objectives • Use inductive reasoning to identify patterns and make conjectures. • Find counterexamples to disprove conjectures. Key Terms • Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. • A conjecture is a statement believed to be true based on inductive reasoning. • A counterexample is a specific example that shows a statement of conjecture is false. Example 1 Identifying a Pattern A pattern in a list of items is a description of what the items have in common. To identify a pattern in a given list, first determine something that the first two items have in common. Then, consider whether the third item also has this commonality. If the commonality holds for each of the items in the list, then the pattern has been identified.

In this example, three lists are given. For each list, describe the pattern and find the next item in each list. Each item in the first list is a letter. So, the next item in the list must also be a letter. Since the first item is the letter A, the pattern might be simply the letters in the alphabet. In this case the second item in the list must be the letter B. However, the second item is E, not B. So, determine what, besides the fact that the items are letters in the alphabet, the items in the list have in common. The letters A and E are the first two vowels in the alphabet, so that is a possible pattern. And since the next item, I, is the third vowel in the alphabet, this pattern holds. So, the next item in the list must be the vowel that follows I in the alphabet, which is the letter O. The second list is composed of numbers. Determine what the numbers have in common to identify the pattern in the list. Each number in the list is a multiple of 7; the first item is 7(1), the second is 7(2), the third is 7(3), and the fourth is 7(4). So, the next item in the list must be 7(5), or 35, since it is the fifth item in the list. The third list is composed of circles and each subsequent circle is divided into a greater number of pieces. The number of parts in each circle is twice the number of parts in the previous circle. In other words, the number of parts is doubled in each item. The third item has 4 parts. Therefore, the fourth item must have 8 parts, since 4(2) = 8.

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