Remember the trees in the Recognize and Describe Numerical Patterns by Finding a Rule Concept? Well, Kelly wrote down a pattern of numbers and we figured out a rule for the pattern. Let's look at the pattern again and the rule for the pattern.

**1, 1, 2, 3, 5, 8, 13....**

This pattern has a rule. This rule is that the two previous numbers add together to equal the next number. Given this information, what is the next number in the pattern?

**When you extend number patterns, you can use the rule to figure out the next numbers in the sequence. In this Concept you will learn how to do this, then you can figure out the next few values in this pattern.**

### Guidance

Once you have figured out a pattern rule it is easy to use that rule to extend the pattern. Extending a pattern involves writing the numbers that come next in the pattern according to the rule.

Let’s look at an example.

**Example**

Find the next term in the following pattern: 3, 6, 9, 12, ____

**First, notice that this is an ascending pattern meaning that it will involve addition, multiplication or both.**

**What is the relationship between these numbers? How were they increased?**

If you think about this question, you can see that each number was increased by adding three.

**To extend the pattern, we simply add three to the last number in the sequence.**

\begin{align*}12 + 3 = 15\end{align*}

**Our answer is 15.**

**Sometimes, you need to extend rule by looking far out into the future.** Let’s look at an example of this.

What is the seventh number in the sequence: 1, 3, 9, 27, ____ ?

**First, let’s figure out the rule.** This is an ascending sequence so it uses addition, multiplication or both. The rule in this case is \begin{align*}\times \ 3\end{align*}.

**Next we can write out the sequence until we get to the seventh number.**

**1, 3, 9, 27, 81, 243, 729, 2187**

**Our answer is 2187.**

Now it's time for you to try a few on your own.

#### Example A

9, 17, 33, ___, ___

**Solution: 65, 129**

#### Example B

3, 10, 31, ___, ___

**Solution: 94, 283**

#### Example C

4, 17, 56, ____, ____

**Solution: 173, 524**

Now let's go back to the trees.

**You just finished learning all about patterns. What is the rule for the Fibonacci pattern of numbers that Sara and Kelly are using?**

**1, 1, 2, 3, 5, 8, 13,**

**If you look you can see that the two previous numbers add together to equal the next number. This is the rule. Given this information, what is the next number in the pattern?**

\begin{align*}8 + 13 = 21\end{align*}

**What is the next one after that?**

\begin{align*}13 + 21 = 34\end{align*}

**By continuing to use this rule, you can continue to extend the numerical pattern.**

### Vocabulary

- Pattern
- a sequence of number or geometric figures that repeats according to a pattern unit or a rule.

- Algebraic Thinking
- thinking in a mathematical way.

- Numerical Patterns
- number patterns that are organized in a sequence according to a rule.

- Ascending Pattern
- a pattern that increases.

- Descending Pattern
- a pattern that decreases.

- Variable
- a letter used to represent an unknown quantity.

- Expression
- combines variables, numbers and operations but does not equal anything because the variables can have different values.

### Guided Practice

Here is one for you to try on your own.

Extend the following pattern.

24, 14, 9, ____, ____

**Answer**

To figure out this rule, we have to examine the operations done to each value to get to the next value.

\begin{align*}24\div 2+2=14\end{align*}

\begin{align*}14\div 2+2=9\end{align*}

The pattern rule is \begin{align*}\div 2 + 2\end{align*}.

\begin{align*}9\div 2 + 2=6.5\end{align*}

\begin{align*}6.5\div 2 + 2=5.25\end{align*}

The next two values are 6.5 and 5.25.

**This is our answer.**

### Video Review

Khan Academy Patterns in Sequences 2

Khan Academy Patterns in Sequences 2

### Practice

Directions: Use what you have learned to extend each numerical pattern.

1. 2, 3, 4, 5, ____, _____

2. 2, 4, 6, 8, _____, _____

3. 2, 5, 11, 23, _____,_____

4. 3, 6, 9, _____, _____

5. 16, 4, _____

6. 3, 8, 18, _____, _____

7. 100, 50, _____, _____

8. 10, 20, 30, 40, _____,_____

9. 15, 30, 45, _____,_____

10. 100, 112, 124, _____, _____

11. 12, 4, 18, 6, 21, _____

12. 40, 4, 120, 12, 130, _____

13. 2.5, _____, 7, 14, 8, 16

14. 25, 12.5, _____

15. 3, 4.5, 6, 7.5, 9, _____