A. Conjectures and Inductive Reasoning

In the Warm Up activity, you used information about previous events to make predictions about future events. In logic, a statement that has not been proven, but appears to be true based on available evidence, is called a conjecture. Your predictions were conjectures and the type of thinking you used to make those conjectures is called inductive reasoning. Inductive reasoning allows you to draw general conclusions based on patterns or examples.

The sequences shown in the following three examples were made using a pattern. In each of the examples, a conjecture is made about the next value in the sequence.

Example 1

Make a conjecture about the missing number in the sequence 3, 6, 12, 24, 48, ____.

Your goal is to find a pattern that applies to all of the values given and to use that pattern to predict the next value. Patterns are not always obvious so it may take some experimenting to find one rule that works.

Step 1
Look for a pattern.

A list of what you know and don't know may help. Sometimes it helps to look at relationships between particular pairs of numbers.

  • Each number is divisible by 3.
  • The values are increasing.
  • The difference between terms is not always the same.
  • 3 + 3 = 6, but 6 + 3 = 9.
  • 3 × 2 = 6 and 6 × 2 = 12.

It looks like the pattern was made by multiplying each value by 2 to determine the next value.

Step 2
Check that the pattern works for all values.

12 × 2 = 24 and 24 × 2 = 48. The pattern works for all of the data.

Step 3
Use the pattern to predict the next value in the sequence.

48 × 2 = 96

Example 2

Make a conjecture about the missing number in the sequence 1, 4, 9, 16, 25, ____.

Step 1
Look for a pattern.

  • The values are increasing.
  • 1 = 12 , 4 = 22, and 9 = 32.

It looks like the pattern was made by squaring consecutive natural numbers.

Step 2
Check that the pattern works for all values.

16 = 42 and 25 = 52

The pattern works for all of the data.

Step 3
Use the pattern to predict the next value in the sequence.

62 = 36

Example 3

Make a conjecture about the missing number in the sequence 1, 1, 2, 3, 5, 8, ____.

Step 1
Look for a pattern.

  • The values are not always increasing.
  • 1 occurs twice
  • 1 + 0 = 1 and 1 × 1 = 1, but neither of these facts will help produce the sequence.
  • The difference between terms is 0, 1, 1, 2, 3 which has some similarity to the given sequence but may not be enough to work with.
  • 1 + 1 = 2 and 1 + 2 = 3

It looks like the pattern was made by adding the previous two numbers.

Step 2
Check that the pattern works for all values.

2 + 3 = 5 and 3 + 5 = 8

The pattern works for all of the data.

Step 3
Use the pattern to predict the next value in the sequence.

5 + 8 = 13