Example 2 Making a Conjecture Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. Inductive reasoning was used in the previous example to determine the next item in each list. When a statement is believed to be based on inductive reasoning, that statement is called a conjecture. 2.1.1 Using Inductive Reasoning to Make Conjectures (continued)
The given statement is a conjecture. So, inductive reasoning will be used to complete each conjecture. Using inductive reasoning means to consider specific examples. So, consider some specific examples that correspond to the given part of the conjecture and then use the pattern from those examples to complete the conjecture. This conjecture begins with “The product of two odd numbers.” So, each example should show the product of any two odd numbers. Three possible products are listed here. In each case, the result is an odd number. Therefore, based on the result of these examples, the complete conjecture will be “The product of two odd numbers is an odd number.” In this example, the conjecture begins with “The number of circle parts formed by n diameters.” So, each example should show a circle with some number of diameters. When choosing the examples, it is often easiest to begin with a small number and increase by 1 in each example. So, the first example shows a circle with 1 diameter, which divides the circle into 2 parts. The second example shows a circle with 2 diameters. This circle is divided into 4 parts. Based on these two examples, the pattern appears to be that the number of parts is equal to twice the number of diameters. Test this result with the third example. In the third example, there are 3 diameters and the circle is divided into 6 parts. Since 6 = 3(2), the pattern “the number of parts is equal to twice the number of diameters” holds. Now apply this pattern to the case of n diameters. If the number of diameters is n , then the number of parts is 2 n . Therefore, the complete conjecture is “The number of circle parts formed by n diameters is 2 n .”
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