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# Inductive Reasoning from Patterns

## Making conclusions based upon observations and patterns.

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Inductive Reasoning from Patterns

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 This

Watch the first two parts of this video.

### Guidance

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.

#### Example A

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

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

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

#### Example B

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

There would be 10 squares in the 10th\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 10+10+2=22\begin{align*}10 +10 + 2 = 22\end{align*} triangles in all.

#### Example C

Look at the pattern 2, 4, 6, 8, 10, \begin{align*}\ldots\end{align*} What is the 19th\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 19th\begin{align*}19^{th}\end{align*} term, but that could take a while. Notice that the 1st\begin{align*}1^{st}\end{align*} term is 21\begin{align*}2 \cdot 1\end{align*}, the 2nd\begin{align*}2^{nd}\end{align*} term is 22\begin{align*}2 \cdot 2\end{align*}, the 3rd\begin{align*}3^{rd}\end{align*} term is 23\begin{align*}2 \cdot 3\end{align*}, and so on. So, the 19th\begin{align*}19^{th}\end{align*} term would be 219\begin{align*}2 \cdot 19\end{align*} or 38.

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### Guided Practice

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, \begin{align*}\ldots\end{align*} What is the 34th\begin{align*}34^{th}\end{align*} term in the pattern?

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

a) What is the next term in the pattern?

b) What is the 10th\begin{align*}10^{th}\end{align*} term?

4. Find the 8th\begin{align*}8^{th}\end{align*} term in the list of numbers: 2,34,49,516,625\begin{align*}2,\frac{3}{4},\frac{4}{9},\frac{5}{16},\frac{6}{25}\ldots\end{align*}

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 34th\begin{align*}34^{th}\end{align*} term would be 342\begin{align*}34 \cdot 2\end{align*} minus 1, which is 67.

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

Therefore, the next term will be 482\begin{align*}48 \cdot 2\end{align*} or 96. To find the 10th\begin{align*}10^{th}\end{align*} term, continue to multiply by 2, or 322222222229=1536\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*}.

4. First, change 2 into a fraction, or 21\begin{align*}\frac{2}{1}\end{align*}. So, the pattern is now 21,34,49,516,625\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 8th\begin{align*}8^{th}\end{align*} term’s numerator is 9. The denominators are the square numbers, so the 8th\begin{align*}8^{th}\end{align*} term’s denominator is 82\begin{align*}8^2\end{align*} or 64. The 8th\begin{align*}8^{th}\end{align*} term is 964\begin{align*}\frac{9}{64}\end{align*}.

### Explore More

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

1. Use the pattern below to answer the questions.
1. Draw the next figure in the pattern.
2. How does the number of points in each star relate to the figure number?
2. Use the pattern below to answer the questions. All the triangles are equilateral triangles.
1. Draw the next figure in the pattern. How many triangles does it have?
2. Determine how many triangles are in the 24th\begin{align*}24^{th}\end{align*} figure.

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

1. 5, 8, 11, 14, 17, \begin{align*}\ldots\end{align*}
2. 6, 1, -4, -9, -14, \begin{align*}\ldots\end{align*}
3. 2, 4, 8, 16, 32, \begin{align*}\ldots\end{align*}
4. 67, 56, 45, 34, 23, \begin{align*}\ldots\end{align*}
5. 9, -4, 6, -8, 3, \begin{align*}\ldots\end{align*}
6. 12,23,34,45,56\begin{align*}\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6} \ldots\end{align*}
7. 23,47,611,815,1019,\begin{align*}\frac{2}{3},\frac{4}{7},\frac{6}{11},\frac{8}{15},\frac{10}{19}, \ldots\end{align*}
8. -1, 5, -9, 13, -17, \begin{align*}\ldots\end{align*}

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

1. 3, 6, 11, 18, 27, \begin{align*}\ldots\end{align*}
2. 3, 8, 15, 24, 35, \begin{align*}\ldots\end{align*}
3. 1, 8, 27, 64, 125, \begin{align*}\ldots\end{align*}
4. 1, 1, 2, 3, 5, \begin{align*}\ldots\end{align*}

### Vocabulary Language: English Spanish

Inductive Reasoning

Inductive Reasoning

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

Equilateral Triangle

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