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)
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 with the specific rule is called a number pattern.
Example: \(1,3,5,7,..... \\ 2,4,6,8,.....\\ 2,4,8,16,32,.....\)
In this type of pattern, there can be two changing features. First, the number of shapes or objects and types of shapes.
Example:
A list of numbers that follows a certain sequence with the specific rule is called a number pattern.
Generally, we come across many 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)
Triangular numbers are the sum of successive counting number starting from \(1.\)
\(1,\;3,\;6,\;10,\;15,\,...\)
\(\to\) It can be defined by the following rule:
\(x_n=\dfrac{n(n+1)}{2}\)
Here, \(x_n=\) value of \(n^{th}\) term
\(n=\) term which we want to find out
Example: Find out the \(15^{th}\) term of the triangular number series.
We know triangular number series shows the sum of successive counting numbers starting from \(1.\)
To find out the \(15^{th}\) term of the triangular number series, we use the following rule:
\(x_n=\dfrac{n(n+1)}{2}\)
Here, \(n=15\)
\(x_{15}=\dfrac{15(15+1)}{2}\)
\(=120\)
Thus, the \(15^{th}\) term of the series is \(120\).
Square numbers are the squares of successive counting numbers-
\((1)^2,\;(2)^2,\;(3)^2,\;(4)^2\,...\)
It can be defined by the following rule:
\(x_n=n^2\)
Here, \(x_n=\) value of \(n^{th}\) term; \(n=\) term which we want to find out.
Example: Find the \(7^{th}\) term of the square numbers series.
We know square number pattern shows the squares of successive counting numbers.
\(x_n=n^2\)
Here, \(n=7\)
\(\therefore\;7^{th}\) term of the square number pattern is:
\(x_7=(7)^2=7×7=49\)
Square numbers are the squares of successive counting numbers-
\((1)^2,\;(2)^2,\;(3)^2,\;(4)^2\,...\)
It can be defined by the following rule:
\(x_n=n^2\)
Here, \(x_n=\) value of \(n^{th}\) term; \(n=\) term which we want to find out.
Example: Find the \(7^{th}\) term of the square numbers series.
We know square number pattern shows the squares of successive counting numbers.
\(x_n=n^2\)
Here, \(n=7\)
\(\therefore\;7^{th}\) term of the square number pattern is:
\(x_7=(7)^2=7×7=49\)
Let's consider an example:
\(1,\;6,\;21,\;—,\;201\)
\(1×6=6\\ 6×6=36\)
\(\to\) Now, try by multiplying with \(3.\)
\((1×3)=3\\ (6×3)=18\)
The product obtained are \(3\) less than the corresponding terms of the given series.
\(\therefore\) We add \(3\) to the products obtained to get the exact terms of the series.
\((1×3)+3=6\\ (6×3)+3=21\)
Thus, the pattern's rule is : Multiply by \(3\) and then add \(3.\)
Let \(x=\) any term of the pattern. So, the rule is: \(3x+3\)
Now, extend the pattern according to the rule.
\((21×3)+3=66\\ (66×3)+3=201\)
Thus, the missing term is \(66\).
A \(248\)
B \(184\)
C \(284\)
D \(148\)
A sequence of repetitive geometric figures or changing shapes or changing directions, with a specific rule is called a geometric pattern.
For example:
In the given pattern, we can see multiple geometric figures are repeating in groups. Two triangles are followed by a circle which is then followed by a rectangle.
A Sequence of triangles rotating in anti-clockwise direction.
B Sequence of triangles rotating in clockwise direction.
C Not following any sequence.
D Sequence of squares rotating in clockwise direction.
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 \(x+4\)
\(\to\) We can write a rule for the Fibonacci series.
\(x_n=x_{n-1}+x_{n-2}\)
Here, \(x_n=\) term in the series
\(x_{n-1}=\) preceding the first number
\(x_{n-2}=\) preceding the second number
A \(1,\;2,\;3,\;5,\;9\)
B \(0,\;1,\;1,\;2,\;3,\;5\)
C \(1,\;3,\;5,\;7\)
D \(1,\;1,\;2,\;2,\;3\)
Step 1: Analyze the geometric pattern and write it's sequence as a number series.
Step 2: Now write a rule for this series like a number pattern.
\(\to\) Here, each term has a different number of units (squares).
\(\to\) Write it's sequence as a number series.
i.e., \(1,\;3,\;5,\;7\,,...\)
\(\to\) So, the pattern can be written as;
\(1+0,\;1+2,\;1+4,\;1+6,\,...\)
\(\to\) So, the rule can be : \(1+2n\).
\(\to\) Let's check for the above pattern,
\(1+2(1)=1+2=3\)
Which is not the \(1^{st}\) term.
Try with another rule,
\(1+2(n-1)\)
\(\Rightarrow\;1+2n-2\)
\(\Rightarrow\;2n-1\)
Let's check the pattern with this rule,
\(2(1)-1=2-1=1\\ 2(2)-1=4-1=3\\ 2(3)-1=6-1=5\)
Thus, the rule for the given geometric pattern is \((2n-1)\)
where \(n\) represents the term which is to be found out.
For example: \(1,\;3,\;5,\;7\,...\) is a number pattern that follows a specific rule by of \('+2'\) i.e. addition operation.
For example: \(1,\;4,\;19,\;94\,...\)
\(\to\) This number pattern is not following a specific rule of single operation.
(1) Multiplication with addition:
(2) Multiplication with subtraction:
(3) Division with addition:
(4) Division with subtraction:
\(1,\;4,\;19,\;94\,...\)
\(\to\) Since, this is an increasing-number pattern therefore, multiplication with addition or multiplication with subtraction combination should work.
\(\to\) When we multiply each term by \(5\) we get,
\(\begin{array}c 1×5&=&5\\ 4×5&=&20\\ 19×5&=&95\\ \end{array}\)
and so on.
\(\to\) We can observe that products obtained are \(1\) more than the corresponding terms of the given series.
Thus, multiplication with subtraction combination is to be used.
\(\therefore\) Subtract \(1\) from each term which is obtained by multiplying by \(5.\)
\(\begin{array}c (1×5)-1&=&5-1&=&4\\ (4×5)-1&=&20-1&=&19\\ (19×5)-1&=&95-1&=&94\\ \end{array}\)
and so on.
\((94×5)-1=470-1=469\)
A \(51\)
B \(50\)
C \(52\)
D \(53\)
When the consecutive terms in a sequence increase by a small constant value, we can use the addition rule to find more terms in that pattern.
Ex. \(1, \;3, \;5, \;7, \;9,......\)
When the consecutive terms in a sequence decrease by a small constant value, we can use the subtraction rule to find more terms in that pattern.
Ex. \(16, \;14, \;12, \;10, ...\)
When the increase in consecutive terms of a sequence is in multiples, we can use the multiplication rule to find more terms in that pattern.
Ex. \(3, \;6, \;12, \;24, ...\)
When the decrease in consecutive terms of a sequence is in multiples, we can use the division rule to find more terms in that pattern.
Ex. \(120, \;60, \;30, \; ...\)
A \(7\)
B \(8\)
C \(9\)
D \(10\)