Sequences & Patterns
Sequences and patterns help us understand how values are arranged and how they change step by step. They train the brain to recognize logic, order, and predictability, which is essential in mathematics, exams, and real life.
In this lesson, you will learn different types of sequences, how to identify patterns, and how to predict missing or future values.
What Is a Pattern?
A pattern is a repeated or regular arrangement of numbers, shapes, or objects that follows a rule.
Patterns help us predict what comes next without listing everything again.
Example: 2, 4, 6, 8, …
Why Patterns Are Important
Patterns help us reduce complexity by understanding rules instead of memorizing values. They appear in nature, daily routines, exams, and logical reasoning problems.
Once the rule is known, future values become easy to find.
What Is a Sequence?
A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in a sequence is called a term.
Sequences are usually written with commas separating the terms.
Example: 5, 10, 15, 20, …
Identifying a Pattern in a Sequence
To identify a pattern, observe how each term changes from the previous one.
Look for addition, subtraction, multiplication, or division.
Example: 3, 6, 9, 12 → add 3 each time
Arithmetic Sequences
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
This constant difference is called the common difference.
Example: 2, 5, 8, 11, …
Finding the Next Term in an Arithmetic Sequence
Once the common difference is known, finding the next term is straightforward.
Example: 7, 10, 13, 16
Common difference = 3 → Next term = 16 + 3 = 19
Geometric Sequences
A geometric sequence is a sequence where each term is multiplied or divided by the same value.
This constant value is called the common ratio.
Example: 2, 4, 8, 16, …
Finding the Next Term in a Geometric Sequence
Multiply the last term by the common ratio to get the next value.
Example: 3, 6, 12
Common ratio = 2 → Next term = 24
Special Number Sequences
Some sequences appear frequently in exams and daily logic problems. Recognizing them quickly saves time.
- Even numbers: 2, 4, 6, 8, …
- Odd numbers: 1, 3, 5, 7, …
- Squares: 1, 4, 9, 16, …
- Cubes: 1, 8, 27, 64, …
Patterns Using Shapes and Objects
Patterns are not limited to numbers. They can be visual, repeating shapes or colors.
These patterns strengthen logical reasoning, especially for school-level and aptitude exams.
Missing Number Patterns
In many problems, one or more terms are missing. The goal is to identify the rule and fill the gap.
Example: 4, 9, ?, 19
Rule: add 5 → Missing number = 14
Sequences & Patterns in Real Life
Patterns are everywhere around us.
- Calendar dates
- Timetables
- Music rhythms
- Construction designs
Recognizing patterns improves planning and prediction.
Sequences & Patterns in Competitive Exams
Almost all competitive exams include:
- Number series questions
- Missing term problems
- Logic-based pattern recognition
Speed comes from practice and pattern familiarity.
Common Mistakes to Avoid
Most mistakes occur due to rushing.
- Assuming addition when multiplication is used
- Ignoring alternating patterns
- Not checking differences carefully
Slow observation saves time later.
Practice Questions
Q1. Find the next term: 5, 10, 15, 20
Q2. Find the next term: 3, 6, 12, 24
Q3. Find the missing number: 1, 4, 9, ?, 25
Quick Quiz
Q1. What is the common difference in 10, 20, 30?
Q2. Which sequence uses multiplication?
Quick Recap
- Patterns follow a rule
- Sequences are ordered lists
- Arithmetic sequences use addition
- Geometric sequences use multiplication
- Used in exams, life, and logic