Number Sequences: Formation and rules

Number Sequences:

What is a Number Sequence?

A number sequence is a list of numbers that are formed by a rule.

Let us explore with an example:

Amile writes the number sequence 3, 9, 15, 21, …..

Do you guess what the next number is in her number sequence?

What are the next 4 numbers in the number sequence given below?

65, 62, 59, 56, 53,………

Let us discuss the rules for writing the number sequences. 

Addition and subtraction patterns:

A number sequence can be made by adding/subtracting the same value each time.

Look at the number line starting from 1 and adding 4 each time.

As we will be adding, we will move to the right side of the number line.

We get the number sequence: 1, 5, 9, 13, ………….

Look at the number line starting from 15 and subtracting 3 each time.

As we will be subtracting, we will move the left side of the number line.

We get the number sequence: 15, 12, 9, 6, 3, 0………

Multiplication and division patterns:

A number sequence can be made by multiplying/dividing by the same value each time.

Let us see how to get the number sequence by multiplying 3 each time.

We get the number sequence: 3, 9, 27, 81, ….

Let us see how to get the number sequence by dividing by 2 each time.

We get the number sequence: 160, 80, 40, 20, 10, ….

How Can You Use a Rule to Continue a Pattern?

Rule: A rule is a mathematical phrase that tells how numbers or shapes in a pattern are related.

Example 1: The stall numbers in an exhibition follow the rule ‘add 6’. If the pattern continues, what are the next four stall numbers? Describe a feature of the pattern.

Using a Number Line to Continue the Pattern::

The rule for the stall is “add 6”.

The next four stall numbers are 10, 16, 22, and 28.

Describe a Feature of the Pattern:

In this pattern, all the stall numbers are even numbers.

Example 2: In another exhibition, the stall numbers follow the rule “subtract 5″. What are the next three stall numbers after 25? Describe a feature of the pattern.

The next three house numbers are 20, 15 and 10. 

Feature of the pattern: All of the stall numbers are multiples of 5.

Some patterns have rules using addition, while others have rules using subtraction.

Example: John’s rule is “add 3”. He started with 3 and wrote the numbers below. Which number does NOT belong to John’s pattern? Explain.

3, 6, 9, 11, 15, 18…….

Solution: John’s rule is “add 3”.

As per John’s rule, the pattern is 3, 6, 9, 12, 15, 18…….

The feature of the pattern is ‘multiples of 3’.

But the given pattern in the question is 3, 6, 9, 11, 15, 18…….

Here, 11 does not belong to John’s pattern. Because, if you add 3 to the previous number 9, we will get 12. It is not in the given pattern.

Problem-Solving:

Problem 1: Look at the rules and starting numbers below. Write the next 4 numbers in each pattern.

Solution: 

Here, the starting number is 15, and the rule is to add 5.

So, start from 15, add 5 each time, and we will get, 

Here, the starting number is 60, and the rule is to subtract 6.

So, start from 60, subtract 6 each time, and we will get,

Problem 2: Look at the rules and starting numbers below. Write the next 4 numbers in each pattern.

Solution: 

Here, the starting number is 2, and the rule is to multiply by 4.  

So, start from 2, multiply by 4 each time, and we will get, Problem 2So, start from 2, multiply by 4 each time, and we will get, 

8, 32, 64, ……………

Here, the starting number is 243, and the rule is to divide by 3.

So, start from 243, divide by 3 each time, and we will get,

81, 27, 9, 3.

Exercise:

1. Subtract 3: 63, 60, 57, ____, _____.
2. Add 7: 444, 451, 458, ____, _____.
   For 3-4 use the rule to generate each pattern.
3. Rule: Subtract 10
4. Rule: Add 51
5. Some patterns use both addition and subtraction in their rules. The rule is “add 3, subtract 2.” Find the next three numbers in the pattern.
   1, 4, 2, 5, 3, 6, 4, 7, ____, _____, _____.

What Have We Learned:

• A number sequence can be made by adding/subtracting the same value each time.
• A number sequence can be made by multiplying/dividing by the same value each time.
• Rule: A rule is a mathematical phrase that tells how numbers or shapes in a pattern are related.