1. Introduction
A sequence is simply an ordered list of numbers written in a specific pattern. Each number in the sequence is called a term. The order in which the numbers appear is important—they follow some rule or pattern.
Examples of sequences we see in daily life:
- Roll numbers in a class (1, 2, 3, 4, ...)
- Even numbers (2, 4, 6, 8, ...)
- Stair steps increasing by the same height
A sequence helps us identify patterns so we can find missing terms or extend the list further.
2. What Exactly Is a Sequence?
A sequence is a list of numbers arranged in a specific order. We usually write it as:
\(a_1, a_2, a_3, a_4, ...\)
Here:
- \(a_1\) is the first term
- \(a_2\) is the second term
- \(a_3\) is the third term
Every term follows a rule. For example, in the sequence 5, 10, 15, 20, ... each term increases by 5.
2.1. Types of Sequences (Basic School Level)
- Increasing sequence: terms keep getting bigger (3, 7, 11, 15, ...)
- Decreasing sequence: terms keep getting smaller (20, 15, 10, 5, ...)
- Constant sequence: all terms are the same (4, 4, 4, 4, ...)
- Finite sequence: limited number of terms
- Infinite sequence: goes on forever
3. What Is a Series?
A series is what we get when we add the terms of a sequence. It is written using plus signs.
Example:
Sequence → 2, 4, 6, 8
Series → 2 + 4 + 6 + 8
A sequence lists numbers. A series adds them.
4. Difference Between Sequence and Series
This small table makes the difference clear:
| Sequence | Series |
|---|---|
| Ordered list of numbers | Addition of numbers in a sequence |
| Uses commas | Uses plus signs |
| Example: 3, 6, 9 | Example: 3 + 6 + 9 |
5. Real-Life Examples of Sequences
Sequences appear in many places around us:
- Number of students increasing class by class
- Marks scored in tests following a pattern
- Steps of a staircase rising evenly
- Page numbers in a book
- Calendar dates
Recognising patterns is important because it helps us predict the next term and understand the behaviour of the entire list.
6. Examples to Understand Sequences
- 1, 4, 7, 10, ... (increasing by 3)
- 20, 18, 16, 14, ... (decreasing by 2)
- 5, 5, 5, 5, ... (constant)
- 2, 4, 8, 16, ... (doubling each time)
7. Common Mistakes
- Thinking a sequence must always increase (it can decrease too).
- Focusing only on the first two terms and assuming the pattern.
- Mixing up a sequence (comma-separated) and a series (plus signs).
- Ignoring the rule that generates the sequence.
8. Quick Practice
Try identifying the pattern in these sequences:
- 3, 6, 9, 12, ...
- 50, 47, 44, 41, ...
- 10, 20, 40, 80, ...
- 5, 5, 5, 5, ...
9. Summary
- A sequence is an ordered list of numbers following a pattern.
- A series is the sum of numbers in a sequence.
- Sequences can increase, decrease, or stay constant.
- Recognising patterns makes it easier to analyse future topics like AP.