What if you were given a pattern of three numbers or shapes and asked to determine the sixth number or shape that fit that pattern? After completing this Concept, you'll be able to use inductive reasoning to draw conclusions like this based on examples and patterns provided.
Watch the first two parts of this video.
One type of reasoning is inductive reasoning. Inductive reasoning entails making conclusions based upon examples and patterns. Visual patterns and number patterns provide good examples of inductive reasoning. Let’s look at some patterns to get a feel for what inductive reasoning is.
A dot pattern is shown below. How many dots would there be in the
Draw a picture. Counting the dots, there are
How many triangles would be in the
There would be 10 squares in the
Look at the pattern 2, 4, 6, 8, 10,
Each term is 2 more than the previous term.
You could count out the pattern until the
1. For two points, there is one line segment connecting them. For three non-collinear points, there are three segments. For four points, how many line segments can be drawn to connect them? If you add a fifth point, how many line segments can be drawn to connect the five points?
2. Look at the pattern 1, 3, 5, 7, 9, 11,
3. Look at the pattern: 3, 6, 12, 24, 48,
a) What is the next term in the pattern?
b) What is the
4. Find the
1. Draw a picture of each and count the segments.
For 4 points there are 6 line segments and for 5 points there are 10 line segments.
2. The next term would be 13 and continue go up by 2. Comparing this pattern to Example C, each term is one less. So, we can reason that the
3. Each term is multiplied by 2 to get the next term.
Therefore, the next term will be
4. First, change 2 into a fraction, or
For questions 1-3, determine how many dots there would be in the
- Use the pattern below to answer the questions.
- Draw the next figure in the pattern.
- How does the number of points in each star relate to the figure number?
- Use the pattern below to answer the questions. All the triangles are equilateral triangles.
- Draw the next figure in the pattern. How many triangles does it have?
- Determine how many triangles are in the
For questions 6-13, determine: the next three terms in the pattern.
- 5, 8, 11, 14, 17,
- 6, 1, -4, -9, -14,
- 2, 4, 8, 16, 32,
- 67, 56, 45, 34, 23,
- 9, -4, 6, -8, 3,
- -1, 5, -9, 13, -17,
For questions 14-17, determine the next two terms and describe the pattern.
- 3, 6, 11, 18, 27,
- 3, 8, 15, 24, 35,
- 1, 8, 27, 64, 125,
- 1, 1, 2, 3, 5,