Coach Gerald enjoys coaching soccer. This year, his team has earned a spot in a regional tournament. The tournament is a knock out tournament. The losing team is automatically out of the tournament and the winning team continues to the next round. Coach Gerald asks his students how many rounds they would have to win in order to be the last team standing. His team was not sure, so he decided to help them by providing part of the pattern that the winners of each round experience.

32, 16, 8, 4, ______

What is the next number in the pattern that Coach Gerald wrote for his players?

In this concept, you will learn how to use patterns to solve problems.

### Using Patterns to Solve Problems

When solving problems that involve greatest common factors, you can use patterns to help.

Let's look at an example.

Name the next value in the series. 4, 7, 10, 13, ______

First, you need to identify the pattern by looking at the numbers.

4 + 3 = 7; 7 + 3 =10, 10 + 3 = 13

The pattern consists of adding 3 to the previous number, so the next number will be the sum of 13 and 3.

13 + 3 = 16

The answer is 16.

Although the first example involved addition, any of the four operations can be used when trying to identify the pattern.

Let's look at an example of a pattern that uses multiplication.

Name the next value in the series. 2, 6, 18, 54, _____

Notice that the next number is found by multiplying the current number by 3. Based on that pattern, the next number will be the product of 54 and 3.

\begin{align*}54\times 3=162\end{align*}

The next number in the series is 162.

### Examples

#### Example 1

Earlier, you were given a problem about Coach Gerald.

His team is in a tournament in which teams are knocked out in each round. He wants his team to understand the number of games that they will have to win in order to win the tournament, so he provides his team with part of the pattern that the winners of each round experience.

32, 16, 8, 4, ______

What is the next number in the pattern that Coach Gerald wrote for his players?

First, figure out what changes between the given values.

32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4

Next, look at the changes and determine the pattern.

Subtract half of the previous number from the current number to calculate the next number.

Then, find the next number.

4 - 2 = 2

The answer is 2.

#### Example 2

What is the next number in this sequence?

1, 1, 2, 3, 5, 8, 13, 21, _____

First, figure out what changes between the given values.

1 + 0 = 1; 1 + 1 = 2; 2 + 1 = 3; 3 + 2 = 5; 5 + 3 = 8; 8 + 5 = 13; 13 + 8 = 21

Next, look at the changes and determine the pattern.

The next number is found by finding the sum of the current number and the number before it.

Then, determine the next number.

21 + 13 = 34

The next number is 34.

#### Example 3

Name the next value in the series.

3, 7, 15, 31, _____

First, figure out what changes between the given values.

3 + 4 = 7; 7 + 8 = 15; 15 + 16 = 31

Next, look at the changes and determine the pattern.

Add 2 times the previous number that was added

Then, determine the next number.

31 + 2(16) = 31 + 32 = 63

The answer is 63.

#### Example 4

Name the next value in the series.

12, 8, 4, _____

First, figure out what changes between the given values.

12 - 4 = 8; 8 - 4 = 4

Next, look at the changes and determine the pattern.

Subtract 4 to calculate the next number

Then, determine the next number.

4 - 4 = 0

The answer is 0.

#### Example 5

Name the next value in the series.

2.5, 5, 10, 20, 40, _____

First, figure out what changes between the given values.

2.5 + 2.5 = 5; 5 + 5 = 10; 10 + 10 = 20; 20 + 20 = 40

Next, look at the changes and determine the pattern.

Add the current number to itself to calculate the next number

Then, determine the next number.

40 + 40 = 80

The answer is 80.

### Review

Figure out the pattern in each of the following problems. Then write in the next number in each pattern.

- 2, 4, 6, 8, 10 ____
- 20, 17, 14, 11, ____
- 4, 8, 16, 32, ____
- 200, 100, 50 ____
- 120, 60, 30, 15, ____
- 22, 33, 44, 55, 66, ____
- 4, 12, 36, _____
- 5, 10, 6, 12, 8, 15, 11, _____
- 6, 4, 8, 5, 10, 6, 12, _____
- 5, 11, 6, 13, 7, 15, 8, _____
- 6, 12, 18, 24, 30, 36, _____
- 100, 75, 100, 75, 50, 75, _____
- 14, 28, 35, _____
- 200, 50, 12.5, _____
- 621, 207, 69, _____

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.21.