# 2.5: Inductive Reasoning from Patterns

- Page ID
- 2138

## 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?

Example \(\PageIndex{1}\)

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 \(\PageIndex{2}\)

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 \(\PageIndex{3}\)

Look at the pattern \(2, 4, 6, 8, 10, \ldots\) 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 \(2 \cdot 1\), the 2nd term is \(2 \cdot 2\), the 3rd term is \(2 \cdot 3\), and so on. So, the \(19^{th}\) term would be \(2 \cdot 19\) or 38.

Example \(\PageIndex{4}\)

Look at the pattern: \(3, 6, 12, 24, 48, \ldots\)

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 \(48 \cdot 2\) 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 \(\PageIndex{5}\)

Find the 8th term in the list of numbers: \(2, \dfrac{3}{4}, \dfrac{4}{9}, \dfrac{5}{16}, \dfrac{6}{25}\ldots\)

**Solution**

First, change 2 into a fraction, or \(\dfrac{2}{1}\). So, the pattern is now \(\dfrac{2}{1}, \dfrac{3}{4}, \dfrac{4}{9}, \dfrac{5}{16}, \dfrac{6}{25}\ldots\) 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 \(\dfrac{9}{64}\).

## 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.

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

For questions 14-17, determine the next two terms **and** describe the pattern.

- \(3, 6, 11, 18, 27, \ldots\)
- \(3, 8, 15, 24, 35, \ldots\)
- \(1, 8, 27, 64, 125, \ldots\)
- \(1, 1, 2, 3, 5, \ldots\)

## Review (Answers)

To see the Review answers, open this PDF file and look for section 2.1.

## 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. |

## Additional Resources

Interactive Element

Video: Inductive Reasoning

Activities: Inductive Reasoning from Patterns Discussion Questions

Study Aids: Types of Reasoning Study Guide

Practice: Inductive Reasoning from Patterns

Real World: The Science of Induction