Pattern in Maths

A pattern is a sequence or design that repeats according to a rule. It can be anything that follows a particular arrangement or order. Patterns can help us make predictions and solve problems more efficiently.

In this article, we are going to learn about the different patterns in math, including their rules and types.

Patterns in Math

In mathematics, patterns refer to a series of numbers, shapes, or objects that follow a predictable sequence or arrangement. These patterns can be found in various mathematical concepts, including arithmetic, algebra, geometry, and even in real-world situations.

For Example:

• Even numbers pattern -: 2, 4, 6, 8, 10, 1, 14, 16, 18, . . .
• Odd numbers pattern -: 3, 5, 7, 9, 11, 13, 15, 17, 19, . . .
• Fibonacci numbers pattern -: 1, 1, 2, 3, 5, 8 ,13, 21, . . .

Rules for Patterns in Math

There are a few rules for patterns in math. These rules help us identify the underlying structure and relationships within patterns, This allows us to extend and generalize our understanding to solve problems and make predictions.

Three of the most common rules for patterns in math are:

• Regularity and Repetition
• Symmetry and Order
• Recursive Rules

Regularity and Repetition

Patterns often show regularity and repetition, where a certain element or group of elements repeats in a predictable manner. This repetition allows us to identify the pattern and extend it further.

• For example, consider the sequence: 2, 4, 6, 8, 10. Here, the regularity of increasing by 2 is evident, indicating an arithmetic sequence. Similarly, in a geometric pattern like a checkerboard, the repetition of black and white squares in a regular alternating pattern demonstrates regularity.

Symmetry and Order

Symmetry and order are common features of patterns, particularly in geometric patterns. Symmetry refers to a balanced arrangement of elements, where one side mirrors the other. This symmetry can be rotational, reflectional, or translational.

• For example, a snowflake exhibits radial symmetry, where its branches are symmetrically arranged around a central point. Also, tessellations demonstrate translational symmetry, where identical shapes repeat in a regular pattern without gaps or overlaps.

Recursive Rules

Some patterns follow recursive rules, where each element is defined in terms of one or more preceding elements. The Fibonacci sequence is an example of a pattern following a recursive rule. In this sequence, each term (starting from the third term) is the sum of the two preceding terms.

• For example, the sequence starts with 0 and 1, and each subsequent term is obtained by adding the two preceding terms: 0, 1, 1, 2, 3, 5, 8, and so on. Recursive rules provide a systematic way to generate new elements in a pattern based on previous elements. This allows for infinite expansion and exploration of patterns.

Types of Patterns

On the basis of their characteristics and the behavior of their elements, patterns in math can be categorized into three groups. They are:

• Repeating patterns
• Growing patterns
• Shrinking patterns

Repeating Patterns

In repeating patterns, the same elements occur in a predictable sequence. For example, in the pattern “red, blue, green, red, blue, green,” colors repeat in a set order.

• These patterns often have a fixed interval or cycle that repeats indefinitely.
• Repeating patterns are common in everyday objects and designs, such as wallpaper.
• They are easy to recognize and extend by identifying the repeating unit.

Growing Patterns

Growing patterns involve elements that increase or decrease in a consistent manner. For example, in the pattern 1, 2, 4, 8, 16, each term doubles the previous one.

• These patterns demonstrate exponential growth or decay over successive terms.
• Growing patterns are often found in sequences involving multiplication or division.
• They can be extended by continuing the same growth or shrinkage rule.

Shrinking Patterns

Shrinking patterns involve elements that decrease in size or quantity over time. For example, in the pattern 100, 50, 25, 12.5, each term halves the previous one.

• These patterns exhibit a reduction or contraction with each successive element.
• Shrinking patterns are common in sequences involving division or subtraction.
• They can be extended by applying the same reduction rule to subsequent terms.

On the basis of the elements they consist of and the rules governing their arrangement, patterns can be divided into three categories. They are:

• Shape patterns
• Letter patterns
• Number patterns

Shape Patterns

Shape patterns involve the repetition or arrangement of geometric shapes. They may include shapes like squares, circles, triangles, or more complex polygons.

• These patterns often exhibit symmetry, rotation, or translation properties.
• Shape patterns can be found in tessellations, mosaics, and architectural designs.
• They are characterized by the visual arrangement of shapes in a repeated sequence.

Letter Patterns

Letter patterns involve the repetition or arrangement of letters from the alphabet. They may include single letters or combinations of letters forming words or phrases.

• These patterns can follow alphabetical order, spelling rules, or specific sequences.
• Letter patterns are commonly found in language-related activities and puzzles.
• They are characterized by the sequential arrangement of letters according to specific rules.

Number Patterns

Number patterns involve the repetition or arrangement of numerical elements. They may include sequences of numbers following arithmetic, geometric, or recursive rules.

• These patterns can exhibit regularity, growth, or shrinkage in successive terms.
• Number patterns are commonly found in mathematics, puzzles, and real-world applications.
• They are characterized by the systematic arrangement of numbers based on mathematical rules.

Number Patterns

Number patterns refer to sequences of numbers that follow a specific rule or pattern. These patterns can be found in various mathematical contexts, from simple arithmetic sequences to more complex geometric or recursive sequences.

The different types of number patterns are algebraic or arithmetic pattern, geometric pattern, Fibonacci pattern, etc. Let us learn three different number patterns here.

Arithmetic Sequence

Arithmetic sequences are sequences of numbers where each term is obtained by adding a constant value to the previous term. This constant value is called the common difference.

General formula for an arithmetic sequence is:

a_n ​= a₁ + (n−1)⋅d

Where:

• a_n is the n-th term,
• a₁ is the first term,
• n is the position of the term in the sequence, and
• d is the common difference.

For example, consider the sequence: 2, 5, 8, 11, 14. Here, the first term, a₁ ​=2 and the common difference, d=3.

Geometric Sequence

Geometric sequences are sequences of numbers where each term is obtained by multiplying the previous term by a constant value. This constant value is called the common ratio.

General formula for a geometric sequence is:

a_n ​=a₁ ​⋅r^(n-1)

Where:

• a_n ​ is the n-th term,
• a₁ ​ is the first term,
• n is the position of the term in the sequence, and
• r is the common ratio.

For example, consider the sequence: 2, 6, 18, 54. Here, the first term, a₁ ​=2 and the common ratio, r = 3.

Fibonacci Sequence

Fibonacci sequence is a special sequence of numbers where each term is the sum of the two preceding terms. It starts with 0 and 1, and the subsequent terms are generated by adding the two previous terms.

Fibonacci Sequence

Fibonacci sequence is represented as:

0, 1, 1, 2, 3, 5, 8, 13, 21, . . .

where each term (starting from the third term) is the sum of the two preceding terms.

Practice Problems On Pattern in Maths

Problem 1: Sequence: 2, 5, 11, 23, ____

What is the next number in the sequence?

Solution:

Solution: The pattern in this sequence involves adding consecutive odd numbers to the previous term. Let’s break it down:

• 2 + 3 = 5
• 5 + 6 = 11
• 11 + 12 = 23

Following this pattern, the next number would be 23 + 13 = 36.

Problem 2: Pattern: Square, Circle, Triangle, Square, Circle, Triangle, ____

What is the next shape in the pattern?

Solution:

The pattern alternates between shapes and progresses in a fixed order. The sequence goes: Square, Circle, Triangle, Square, Circle, Triangle. So, the next shape would be a Square.

Problem 3: Pattern: A, C, F, J, ____

What is the next letter in the sequence?

Solution:

The pattern here involves increasing the gap between consecutive letters: +1, +2, +3, +4, and so on. Starting from A, the next letter is obtained by adding 1, then 2, then 3, and so forth.

• A + 1 = B
• B + 2 = D
• D + 3 = G
• G + 4 = K

Following this pattern, the next letter would be K + 5 = P.

FAQs on Patterns in Maths

What are arithmetic patterns in math?

Arithmetic patterns are sequences of numbers where each term is obtained by adding (or subtracting) a constant value to (or from) the previous term. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

How do geometric patterns differ from arithmetic patterns?

Geometric patterns involve sequences of numbers where each term is obtained by multiplying (or dividing) the previous term by a constant value. Unlike arithmetic patterns, geometric patterns exhibit multiplication or division relationships between terms.

What is the significance of recognizing patterns in mathematics?

Recognizing patterns in mathematics helps in problem-solving, making predictions, and understanding mathematical concepts more deeply. It also facilitates logical thinking and analytical skills development.

Can patterns be found in shapes and figures as well?

Yes, patterns can be found in shapes and figures, known as shape patterns. These patterns involve the repetition or arrangement of geometric shapes, often exhibiting symmetry, rotation, or translation properties.

How are letter patterns useful in language-related activities?

Letter patterns involve the repetition or arrangement of letters from the alphabet, helping in language-related activities such as spelling, vocabulary building, and word formation. They are commonly used in puzzles, games, and educational exercises.

What are some real-world applications of number patterns?

Number patterns are widely used in various real-world applications, including financial analysis, scientific research, engineering, and data analysis. They help in predicting trends, modeling phenomena, and making informed decisions.