### 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 \begin{align*}4^{th}\end{align*} figure? How many dots would be in the \begin{align*}6^{th}\end{align*} figure?

Draw a picture. Counting the dots, there are \begin{align*}4 + 3 + 2 + 1 = 10 \ dots\end{align*}.

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

#### Example 2

How many *triangles* would be in the \begin{align*}10^{th}\end{align*} figure?

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

#### Example 3

Look at the pattern 2, 4, 6, 8, 10, \begin{align*}\ldots\end{align*} What is the \begin{align*}19^{th}\end{align*} term in the pattern?

Each term is 2 more than the previous term.

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

#### Example 4

Look at the pattern: 3, 6, 12, 24, 48, \begin{align*}\ldots\end{align*}

What is the next term in the pattern? What is the \begin{align*}10^{th}\end{align*} term?

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

Therefore, the next term will be \begin{align*}48 \cdot 2\end{align*} or 96.

To find the \begin{align*}10^{th}\end{align*} term, continue to multiply by 2, or \begin{align*}3 \cdot \underbrace{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}_{2^9} = 1536\end{align*}.

#### Example 5

Find the \begin{align*}8^{th}\end{align*} term in the list of numbers: \begin{align*}2,\frac{3}{4},\frac{4}{9},\frac{5}{16},\frac{6}{25}\ldots\end{align*}

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

### Review

For questions 1-3, determine how many dots there would be in the \begin{align*}4^{th}\end{align*} and the \begin{align*}10^{th}\end{align*} 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 \begin{align*}24^{th}\end{align*} figure.

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

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

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

- 3, 6, 11, 18, 27, \begin{align*}\ldots\end{align*}
- 3, 8, 15, 24, 35, \begin{align*}\ldots\end{align*}
- 1, 8, 27, 64, 125, \begin{align*}\ldots\end{align*}
- 1, 1, 2, 3, 5, \begin{align*}\ldots\end{align*}

### Review (Answers)

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

### Resources