Examples:
(1) 1, 3, 5, 7, 9,........
In this example, there is a relation between every next term with the previous term. Every next term is increased by 2.
(2) Consider the given figure.
In this example, there is a sequence of two circles after every triangle.
(i) Number patterns
(ii) Figure patterns
A list of numbers that follows a certain sequence using a specific rule is called number pattern.
Ex. \(1,3,5,7,..... \\ 2,4,6,8,.....\\ 2,4,8,16,32,.....\)
In this type of pattern, there can be two changing features:
(1) Number of shapes or objects
(2) Types of shapes
A list of numbers that follows a certain sequence using a specific rule is called number pattern.
There can be different types of sequences.
For example:
1, 3, 5, 7, 9,....... Odd number series
2, 4, 6, 8, 10,.....Even number series
Multiples Series:
Series of 2's: 2, 4, 6, 8,.......
Series of 3's: 3, 6, 9, 12,.......
Series of 4's: 4, 8, 12, 16,.......
Series of 5's: 5, 10, 15, 20,.......
and so on.
Composite Number Series:
4, 6, 8, 9, 10, 12, 14, 15, 16,......
Prime Number Series:
2, 3, 5, 7, 11, 13, 17, 19, 23,......
Square Numbers:
1, 4, 9, 16, 25, ......
\(\Rightarrow\) \(1^2,\;2^2,\;3^2,\;4^2,\;5^2,...\) (Squares of the counting numbers)
Cube Numbers:
1, 8, 27, 64, 125, ......
\(\Rightarrow\) \(1^3,\;2^3,\;3^3,\;4^3,\;5^3,...\) (Cubes of the counting numbers)
A \(3,\;6,\;9,\;12,....\)
B \(4,\;8,\;12,\;16,....\)
C \(1^3,\;2^3,\;3^3,\;4^3,\;...\)
D \(2,\;4,\;6,\;8,....\)
Consider an example:
\(3, \;6, \;9, \;12,......\)
In this example, we can use a variable to represent the terms in the given sequence.
Let \(x=\) any term in the pattern
\(x\) plus 3 means "\(x\)+3".
\(3, \;6, \;9, \;12,\dots\)
So, the rule is working.
A \(2x\)
B \(3x\)
C \(x+2\)
D \(2x-1\)
In this type of pattern, there can be two changing features, first number of shapes or objects and other is types of shapes.
Consider the given example as shown.
Example 1: In this example, shapes are changing directions.
Example 2:
In this example, the number of shapes is changing.
This can be done by following two steps:
(i) Think of the input as a variable.
(ii) Write the operations used with this variable to get the output.
Consider an example:
Input | Output |
0 | 0 |
1 | 4 |
2 | 8 |
3 | 12 |
Rule\(\rightarrow4x\)
If we substitute \(0\) in place of \(x\), the output is: \(4×0=0\)
If we substitute \(1\) in place of \(x\), the output is: \(4×1=4\)
If we substitute \(2\) in place of \(x\), the output is: \(4×2=8\)
\(\vdots\)
So the rule is working.
The input-output can respectively be written as \(x\) and \(y\) in an input-output table.
A \(x+1\)
B \(x+2\)
C \(x+3\)
D \(x+4\)