# 2.3: Number Patterns

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Identify missing values in numerical patterns.

## Extend Numerical Patterns Figure $$\PageIndex{1}$$

Nicholas is on his way to his cousin's new house. He found the street and is just looking for the house. His cousin lives at number 1644. Nicholas notices that the numbers of the houses on the right side of the street seem to follow a pattern:

$$1574, 1584, 1594, …$$

Number 1574 was the first house Nicholas saw on the right. If he assumes that the pattern of how the houses are numbered will continue, how can Nicholas extend the pattern to figure out how many houses away his cousin's house is?

In this concept, you will learn how to extend numerical patterns.

## Numerical Patterns

A numerical pattern is a sequence of numbers that has been created based on a rule called a pattern rule. Pattern rules can use one or more mathematical operations to describe the relationship between consecutive numbers in the sequence.

Knowing the pattern rule will help you to extend the pattern. To extend the pattern means to use the pattern rule to write the numbers that would come next in the sequence.

Here is an example.

Find the next two numbers in the following sequence: 3, 6, 9, 12,$$\underline{\quad}$$, $$\underline{\quad}$$.

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves addition or multiplication.

Next, come up with potential pattern rules. Think: "What could you do to 3 to get 6?"

• You could multiply by 2.
• You could do a combination of two or more operations.

Now, check if any of these potential pattern rules work with the rest of the sequence.

Consider 6 and 9.

• If you add 3 to 6 you get 9. So the pattern rule "add 3" seems to work.

Make sure "add 3" works throughout the whole sequence.

"Add 3" works for the whole sequence.

Now, extend the pattern. Apply the pattern rule of "add 3" to the 12 at the end of the pattern.

The next number in the pattern will be 15.

"Add 3" one more time to get the sixth number in the sequence.

The answer is that the extended pattern is 3, 6, 9, 12, $$\underline{15}$$, $$\underiline{18}$$.

Sometimes you will be interested in a particular term in the pattern. In order to figure out a particular term, keep extending the pattern until you've reached the term you are looking for.

Let's look at an example.

What is the seventh number in the sequence: $$1,3,9,27, \ldots$$?

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves addition or multiplication.

Next, come up with potential pattern rules. Think: "What could you do to 1 to get 3?"

• You could multiply by 3.
• You could do a combination of two or more operations.

Now, check if any of these potential pattern rules work with the rest of the sequence.

Consider 1 and 3.

• If you add 2 to 3 you get 5, not 9. So the pattern rule is not "add 2."
• If you multiply 3 by 3 you get 9. So the pattern rule "multiply by 3" seems to work.

Make sure "multiply by 3" works throughout the whole sequence.

"Multiply by 3" works for the whole sequence.

Finally, you can extend the pattern to find the seventh number. Keep multiplying by 3 until you reach the seventh number in the sequence.

The first seven numbers in the sequence are $$1, 3, 9, 27, 81, 243, 729$$.

The answer is that the seventh number in the sequence is 729.

Example $$\PageIndex{1}$$

Earlier, you were given a problem about Nicholas, who is going to his cousin's house.

He just passed house number 1574 and noticed that the house numbers seem to follow a pattern:

$$1574, 1584, 1594$$

His cousin lives at number 1644. Nicholas wants to extend the pattern to predict how many houses away his cousin's house is.

Solution

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.

Next, look at how the numbers are related. The difference between the numbers is a constant 10. This means that the pattern rule is "add 10." Double check that this pattern rule works for the three given numbers in the sequence.

The pattern rule "add 10" works for the three given numbers in the sequence.

Now, extend the pattern. Apply the pattern rule "add 10" at the end of the pattern until you hit 1644.

Next, write out the extended sequence.

Notice that 1644 is the eighth number in the sequence.

The answer is that Nicholas can expect his cousin's house to be the eighth house on the right.

Example $$\PageIndex{2}$$

Find the next two numbers in the following sequence: 24, 14, 9, $$\underline{\quad}$$, $$\underline{\quad}$$.

Solution

First, figure out the pattern rule. This is a descending pattern so the rule likely involves division or subtraction.

Next, look at how the numbers are related. The difference between the numbers decreases as you move through the sequence. To get from 24 to 14 you have to subtract 10, but to get from 14 to 9 you have to subtract 5. This means division is involved. Since there is no whole number you can divide 24 by to get 14, the pattern rule is likely division with either addition or subtraction.

Now, consider possible pattern rules that involve division and addition or subtraction. Think: "What could you do to 24 to get 14?"

• You could divide by 2 and add 2.
• You could divide by 3 and add 6.
• You could do some other combination of two or more operations.

Next, look back at the rest of the sequence. Consider 14 and 9.

• If you divide 14 by 2 and add 2 you get 9. So the pattern rule "divide by 2 and add 2" works for the three given numbers in the sequence.

Now, extend the pattern. Apply the pattern rule of "divide by 2 and add 2" at the end of the pattern two times.

The next two numbers in the sequence will be 6.5 and 5.25.

The answer is that the extended pattern is 24, 14, 9, $$\underline{6.5}$$, $$\underline{5.25}$$.

Example $$\PageIndex{3}$$

Find the next two numbers in the following sequence: 9,17,33, $$\underline{\quad}$$, $$\underline{\quad}$$.

Solution

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.

Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 9 by to get 17, the pattern rule is likely multiplication with either addition or subtraction.

Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 9 to get 17?"

• You could multiply by 2 and subtract 1.
• You could multiply by 3 and subtract 10.
• You could do some other combination of two or more operations.

Next, look back at the rest of the sequence. Consider 17 and 33.

• If you multiply 17 by 2 and subtract 1 you get 33. So the pattern rule multiply by 2 and subtract 1 works for the three given numbers in the sequence.

Now, extend the pattern. Apply the pattern rule of "multiply by 2 and subtract 1" at the end of the pattern two times.

The next two numbers in the pattern will be 65 and 129.

The answer is that the extended pattern is 9, 17, 33, $$\underline{65}$$, $$\underline{129}$$.

Example $$\PageIndex{4}$$

Find the next two numbers in the following sequence: 3, 10, 31, $$\underline{\quad}$$, $$\underline{\quad}$$.

Solution

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.

Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 3 by to get 10, the pattern rule is likely multiplication with either addition or subtraction.

Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 3 to get 10?"

• You could multiply by 2 and add 4.
• You could multiply by 3 and add 1.
• You could do some other combination of two or more operations.

Next, look back at the rest of the sequence. Consider 10 and 31.

• If you multiply 10 by 3 and add 1 you get 31. So the pattern rule "multiply by 3 and add 1" works for the three given numbers in the sequence.

Now, extend the pattern. Apply the pattern rule "multiply by 3 and add 1" at the end of the pattern two times.

The next two numbers in the pattern will be 94 and 283.

The answer is that the extended pattern is 3, 10, 31, $$\underline{94}$$, $$\underline{283}$$.

Example $$\PageIndex{5}$$

Find the sixth number in the following sequence: $$4, 17, 56, \ldots$$

Solution

First, figure out the pattern rule. This is an ascending pattern so the rule likely involves multiplication or addition.

Next, look at how the numbers are related. The difference between the numbers increases as you move through the sequence. This means multiplication is involved. Since there is no whole number you can multiply 4 by to get 17, the pattern rule is likely multiplication with either addition or subtraction.

Now, consider possible pattern rules that involve multiplication and addition or subtraction. Think: "What could you do to 4 to get 17?"

• You could multiply by 2 and add 9.
• You could multiply by 3 and add 5.
• You could multiply by 4 and add 1.
• You could do some other combination of two or more operations.

Next, look back at the rest of the sequence. Consider 17 and 56.

• If you multiply 17 by 3 and add 5 you get 56. So the pattern rule "multiply by 3 and add 5" works for the three given numbers in the sequence.

Now, extend the pattern. Apply the pattern rule of "multiply by 3 and add 5" at the end of the pattern three times in order to get to the sixth number in the sequence.

The extended pattern is $$4, 17, 56, 173, 524, 1577$$.

The answer is that the sixth number in the sequence is 1577.

## Review

Extend each numerical pattern by filling in the blanks.

1. 2, 3, 4, 5, $$\underline{\quad}$$, $$\underline{\quad}$$
2. 2, 4, 6, 8, $$\underline{\quad}$$, $$\underline{\quad}$$
3. 2, 5, 11, 23, $$\underline{\quad}$$, $$\underline{\quad}$$
4. 3, 6, 9, $$\underline{\quad}$$, $$\underline{\quad}$$
5. 64, 16, 4, $$\underline{\quad}$$, $$\underline{\quad}$$
6. 150, 100, 50, $$\underline{\quad}$$, $$\underline{\quad}$$
7. 10, 20, 30, 40,$$\underline{\quad}$$, $$\underline{\quad}$$
8. 15, 30, 45,$$\underline{\quad}$$, $$\underline{\quad}$$
9. 100, 112, 124,$$\underline{\quad}$$, $$\underline{\quad}$$
10. 4, 18, 74,$$\underline{\quad}$$, $$\underline{\quad}$$
11. 40, 120, 360,$$\underline{\quad}$$, $$\underline{\quad}$$
12. 2.5, 6, 13,$$\underline{\quad}$$, $$\underline{\quad}$$
13. 50, 25, 12.5,$$\underline{\quad}$$, $$\underline{\quad}$$
14. 3, 4.5, 6, 7.5, 9,$$\underline{\quad}$$