# 7.3.1: Inductive Reasoning from Patterns

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# Inductive Reasoning from Patterns

# Inductive Reasoning

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.

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?

# Examples

Example 1

A dot pattern is shown below. How many dots would there be in the 4^{th} figure? How many dots would be in the 6^{th} figure?

**Solution**

Draw a picture. Counting the dots, there are \(\ 4+3+2+1=10\) dots.

For the 6^{th} figure, we can use the same pattern, \(\ 6+5+4+3+2+1\). There are 21 dots in the 6^{th} figure.

Example 2

How many * triangles* would be in the 10

^{th}figure?

**Solution**

There would be 10 squares in the 10^{th} figure, with a triangle above and below each one. There is also a triangle on each end of the figure. That makes \(\ 10+10+2=22\) triangles in all.

Example 3

Look at the pattern 2, 4, 6, 8, 10, … What is the 19^{th} term in the pattern?

**Solution**

Each term is 2 more than the previous term.

You could count out the pattern until the 19^{th} term, but that could take a while. Notice that the 1^{st} term is 2⋅1, the 2^{nd} term is 2⋅2, the 3^{rd} term is 2⋅3, and so on. So, the 19^{th} term would be 2⋅19 or 38.

Example 4

Look at the pattern: 3, 6, 12, 24, 48, …

What is the next term in the pattern? What is the 10^{th} term?

**Solution**

Each term is * multiplied* by 2 to get the next term.

Therefore, the next term will be 48⋅2 or 96.

To find the 10^{th} term, continue to multiply by 2, or \(\ 3 \cdot \underbrace{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}_{2^{9}}=1536\)

Example 5

Find the 8^{th} term in the list of numbers: \(\ \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \frac{6}{25} \ldots\)

**Solution**

First, change 2 into a fraction, or \(\ \frac{2}{1}\). So, the pattern is now \(\ \frac{2}{1}, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \frac{6}{25} \ldots\) The top is 2, 3, 4, 5, 6. It increases by 1 each time, so the 8^{th} term’s numerator is 9. The denominators are the square numbers, so the 8^{th} term’s denominator is 8^{2} or 64. The 8^{th} term is \(\ \frac{9}{64}\).

# Review

For questions 1-3, determine how many dots there would be in the 4^{th} and the 10^{th} pattern of each figure below.

- 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 24
^{th}figure.

For questions 6-13, determine: the next three terms in the pattern.

- \(\ 5,8,11,14,17, \ldots\)
- \(\ 6, 1, -4, -9, -14, \ldots\)
- \(\ 2, 4, 8, 16, 32, \ldots\)
- \(\ 67, 56, 45, 34, 23, \ldots\)
- \(\ 9, -4, 6, -8, 3, \ldots\)
- \(\ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} \ldots\)
- \(\ \frac{2}{3}, \frac{4}{7}, \frac{6}{11}, \frac{8}{15}, \frac{10}{19}, \ldots\)
- \(\ -1,5,-9,13,-17, \ldots\)

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, …

# Resources

# Vocabulary

Term | Definition |
---|---|

Inductive Reasoning |
Inductive reasoning is a type of reasoning where one draws conclusions from patterns and previous examples. |

Equilateral Triangle |
An equilateral triangle is a triangle in which all three sides are the same length. |