1. Meaning of a Sequence
A sequence is an ordered list of numbers where each number follows a specific idea or pattern. The numbers in the list are called terms. What makes a sequence special is that every term has a fixed position: first term, second term, third term, and so on.
A sequence may grow, shrink, repeat, or follow a rule. The goal is to understand how each term is related to its position in the list.
2. Representation of a Sequence
A sequence can be written or represented in different ways. Each method shows the pattern from a different angle, making it easier to understand or work with.
2.1. Listing Method
In this method, the terms of the sequence are written one after another. For example, the sequence of even numbers can be written as:
\( 2,\ 4,\ 6,\ 8,\ 10,\ldots \)
This method is simple but may not be useful for long or complicated patterns.
2.2. Rule or Formula Method
Here, a formula gives the term in the sequence. If \(a_n\) is the nth term, then a formula tells how to find \(a_n\) for any position \(n\).
Example: The sequence \( 3, 6, 9, 12, 15, \ldots \) can be written using the rule \( a_n = 3n \).
2.3. Recursive Method
A recursive sequence is defined using its previous term. That means each term depends on the term before it.
For example, a sequence where each term is 5 more than the previous term can be written as:
- First term: \( a_1 = 2 \)
- Rule: \( a_n = a_{n-1} + 5 \)
This method focuses on how the sequence grows step by step.
3. Terms of a Sequence
Every number in a sequence is called a term. The position of a term is important because it tells how the term is formed using the pattern.
3.1. General Term \(a_n\)
The general term gives a formula for the term at any position \(n\). Once the formula is known, we can find any term without writing the whole sequence.
Example: If \( a_n = 2n - 1 \), then the 5th term is:
\( a_5 = 2(5) - 1 = 9 \)
3.2. Initial Terms
The first few terms of a sequence often show the pattern clearly. These are called the initial terms. They help identify how the sequence starts and what kind of rule it follows.
Example: In the sequence \( 4, 7, 10, 13, \ldots \), the initial terms show that each number increases by 3.
4. Examples of Simple Sequences
Many common sets of numbers naturally form sequences. These examples help build intuition about patterns and structure.
4.1. Natural Numbers
The natural numbers form the sequence:
\( 1, 2, 3, 4, 5, \ldots \)
This is the most basic and familiar sequence.
4.2. Even and Odd Numbers
Even numbers follow the pattern:
\( 2, 4, 6, 8, 10, \ldots \)
Odd numbers follow the pattern:
\( 1, 3, 5, 7, 9, \ldots \)
Both sequences grow steadily with a constant difference of 2.
4.3. Prime Numbers
Prime numbers form a special sequence because they do not follow a simple formula. However, they follow a clear idea: each number has exactly two factors — 1 and itself.
Prime sequence:
\( 2, 3, 5, 7, 11, 13, 17, \ldots \)