Rules and Relationship Patterns

Patterns with Numbers

​An ordered set of numbers or mathematical entities.

Sequences can be formed by patterns. Patterns are created and arranged following a rule or a set of rules. To solve pattern problems, you must be able to identify the pattern.

Below are methods to help you find the pattern.

​(1) Look at Two Neighboring Terms

​Sequences

​We choose one number in the sequence and look at its two neighboring numbers to see if any rule of addition, subtraction, multiplication, or division applies.

​(1) Look at Two Neighboring Terms

​Example 1. Complete the pattern: 2, 5, 8, 11, _ , 17, 20.

​Solution: 14.

We pick the number 5 and look at its two neighboring numbers 2 and 8. Each number is 3 more than the number before it, so the number after 11 is 11 + 3 = 14.

​(1) Look at Two Neighboring Terms

​Example 2. Complete the pattern: 19, 17, 15, _ , 11, 9.

​Solution: 13.

We pick the number 17 and look at two neighboring numbers 15 and 19. Each number is 2 less than the number before it, so the number after 15 is 15 – 2 = 13.

​(1) Look at Two Neighboring Terms

​Example 3. Complete the pattern: 1, 3, 9, 27, _ , 243.

​Solution: 81.

We pick the number 3, and look at two neighboring numbers 1 and 9. Each number is 3 times the number before it, so the number after 27 is 27 × 3 = 81.

​(1) Look at Two Neighboring Terms

​Example 4. Complete the pattern: 64, 32, 16, 8, _ , 2, 1.

​(2) Finding A Special Relationship

If we are still not able to find the pattern after we examine the relationship of the neighboring terms, we should examine all the terms to see if there are any special rules.

​​(2) Finding A Special Relationship

​Example 5. Sally is listing some counting numbers below. What number goes in the blank?

1, 4, 9, 16, 25, _ , 49, 64

​Solution: 36.

In the sequence, each term is of the form n × n, where n is the order of the term in the sequence. For example, the first term is 1 × 1 = 1, the second term is 2 × 2 = 4, the third term is 3 × 3 = 9, the fourth term is 4 × 4 = 16, and the fifth term is 5 × 5 = 25. So the number in the box, the sixth term, is 6 × 6 = 36, .

​​(2) Finding A Special Relationship

​Example 6. Tommy is listing some counting numbers below. What number goes in the blank?

2, 6, 12, 20, _ , 42.

​Solution: 30.

In the sequence, each term is of the form n × (n + 1), where n is the order of the term in the sequence. For example,

the first term is 1 × 2 = 2,

the second term is 2 × 3 = 6, the third term is 3 × 4 = 12, the fourth term is 4 × 5 = 20,

and the fifth term, the number in the box, will be 5 × 6 = 30.

​​(2) Finding A Special Relationship

​Example 7. What is the fifth term in the following sequence: 2, 12, 30, 56, and so on?

​Solution: 90.

In the sequence, each term is listed as follows:

1 × 2 = 2,

3 × 4 = 12,

5 × 6 = 30.

7 × 8 = 56.

9 × 10 = 90. The fifth term is 90.

(3) Regrouping

In some sequences, the pattern may not be able to be seen by examining two or three neighboring terms, and there is no special relationship that we can see. In these situations, the regrouping method may help.

(3) Regrouping

​Example 8. Complete the pattern: 1, 2, 3, 1, 2, 6, 1, 2, 12, 1, 2, 24, 1, 2, _.

​Solution: 48.

We regroup the terms by 3:

(1, 2, 3)

(1, 2, 6)

(1, 2, 12)

(1, 2, 24)

(1, 2, ?).

If we take the third term from each subgroup, we form the following sequence:

3, 6, 12, 24,…..

Each term is generated by multiplying the previous term by 2.

So our answer is 24 × 2 = 48.

(3) Regrouping

​Example 9. Complete the pattern: 1, 3, 3, 5, 5, 7, _, _.

(3) Regrouping

​Example 10. Find the next two terms in the sequence 8, 1, 10, 2, 12, 3, _ , _.

(3) Regrouping

​Example 11. Find the next two terms in the sequence 15, 6, 13, 7, 11, 8,_ , _.

(3) Regrouping

​Example 12. Find the next term in the sequence 1, 2, 2, 4, 3, 8, 4, 16, 5, _.

(3) Regrouping

​Example 13. Find the next two terms in the sequence 2, 1, 4, 3, 6, 9, 8, 27, 10,_ , _.

(3) Regrouping

​Example 14. The first three groups of a sequence are (1, 5, 9), (2, 10, 18), and (3, 15, 27). Find the sum of the three numbers in the tenth group.

(4). Ignoring Some Terms

Another method to find the pattern of a sequence is to ignore of the terms, commonly, the first term.

(4). Ignoring Some Terms

​Example 15. Find the next term in the sequence 1, 1, 2, 3, 5, 8, 13, 21, _.

​Solution: 34

We ignore the first two terms and instead considering the following sequence: 2, 3, 5, 8, 13, 21, _.

After the second term of our original sequence, each term is the sum of the two proceeding terms. 1 + 1 = 2.

1 + 2 = 3.

2 + 3 = 5.

3 + 5 = 8.

5 + 8 = 13.

8 + 13 = 21.

13 + 21 = 34. The answer is 34.

Note: This is a famous sequence known as the Fibonacci Sequence

(4). Ignoring Some Terms

​Example 16. Tommy is listing some counting numbers below. What number goes in the blank?

1, 3, 4, 7, 11, 18, , 47,….

​Solution: 29

We ignore the first term and consider the following sequence: 3, 4, 7, 11, 18, , 47,….

After the second term of our original sequence, each term is the sum of the two proceeding terms. 1 + 3 = 4.

3 + 4 = 7.

4 + 7 = 11.

7 + 11 = 18.

11 + 18 = 29.

The answer is 29.

(4). Ignoring Some Terms

​Example 17. Find the next term in the sequence 1, 1, 1, 3, 5, 9, 17, _.

​Solution: 31

We ignore the first three terms and consider the following sequence: 3, 5, 9, 17, _.

After the third term of our original sequence, each term is the sum of the two proceeding terms.

1 + 1 + 1 = 2.

1 + 1 + 3 = 5.

1 + 3 + 5 = 9.

3 + 5 + 9 = 17.

5 + 9 + 17 = 31.

The answer is 31.

(4). Ignoring Some Terms

​Example 18. Complete the pattern: 1, 2, 4, 7, 11, 16,_ , _.

​Solution: 22 and 29

We ignore the first term and consider the following sequence: 2, 4, 7, 11, 16, , .

1 + 1 = 2

2 + 2 = 4

4 + 3 = 7

7 + 4 = 11

11 + 5 = 16

16 + 6 = 22,

22 + 7 = 29.

So the answers are 22 and 29.

(4). Ignoring Some Terms

​Example 19. Complete the pattern: 2, 5, 11, 23, 47,_ , _.

​Solution: 95 and 191.

We ignore the first term and consider the following sequence: 5, 11, 23, 47, _,_ .

2 × 2 + 1 = 5

5 × 2 + 1 = 11

11 × 2 + 1 = 23

23 × 2 + 1 = 47

47 × 2 + 1 = 95

95 × 2 + 1 = 191.

So the answers are 95 and 191.

(4). Ignoring Some Terms

​Example 20. Find the next term in the sequence 1, 2, 5, 13, 34, 89, _.

​Solution: 233

After the third term of our original sequence, each term is twice the previous term, plus the remaining previous terms.

2 × 2 + 1 = 5.

5 × 2 + 2 + 1 = 13.

13 × 2 + 5 + 2 + 1 = 34.

34 × 2 + + 34 + 5 + 2 + 1 = 89.

89 × 2 + 34 + 13 + 5 + 2 + 1 = 233.

The answer is 233.