1. Definition of an Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term is obtained by adding or subtracting the same fixed number to the previous term. This fixed number is called the common difference.
Example:
\( 5,\ 8,\ 11,\ 14,\ 17,\ldots \)
Here, every term increases by 3, so the common difference is \( d = 3 \).
2. Common Difference
The common difference is the most important feature of an arithmetic sequence. It tells how the sequence grows or shrinks from one term to the next.
2.1. Meaning of Common Difference \(d\)
The common difference \( d \) is the amount added or subtracted to get from one term to the next.
In general:
\( d = a_{n} - a_{n-1} \)
If \( d > 0 \), the sequence increases. If \( d < 0 \), the sequence decreases.
2.2. Finding \(d\) from Terms
The common difference can be found by subtracting any term from the term that follows it.
Example:
Sequence: \( 12,\ 9,\ 6,\ 3,\ldots \)
Here,
\( d = 9 - 12 = -3 \)
Negative \( d \) means the sequence is decreasing.
3. General Term of an Arithmetic Sequence
The general term gives a formula for finding the term at any position without writing the whole sequence. It is one of the most useful tools for working with arithmetic sequences.
3.1. Formula for \(a_n\)
If \( a_1 \) is the first term and \( d \) is the common difference, then the nth term is:
\( a_n = a_1 + (n - 1)d \)
This formula can be used to jump directly to any term.
3.2. Using \(a_n\) to Find Unknown Terms
The formula can be used backwards as well. If a later term is known, you can find earlier terms or the value of \( d \).
Example:
Given \( a_{12} = 50 \), \( a_1 = 5 \):
\( 50 = 5 + 11d \)
\( 45 = 11d \Rightarrow d = \dfrac{45}{11} \)
4. Finding Terms of an AP
Once the rule or a few terms are known, any missing term can be found using the properties of arithmetic sequences.
4.1. Finding Any Term from Given Terms
If you know two terms, you can find the common difference and then compute any other term.
Example:
If \( a_3 = 14 \) and \( a_7 = 26 \):
Difference in terms = \( 7 - 3 = 4 \)
Difference in values = \( 26 - 14 = 12 \)
So, \( d = \dfrac{12}{4} = 3 \).
Then use \( a_n = a_1 + (n-1)d \) to find any term.
4.2. Inserting Terms Between Two Numbers
To insert terms between two numbers, treat the numbers as the first and last terms of a new arithmetic sequence.
Example:
Insert 3 terms between 2 and 20.
There will be 5 terms total: \( a_1 = 2 \), \( a_5 = 20 \).
Use \( a_5 = a_1 + 4d \):
\( 20 = 2 + 4d \Rightarrow d = \dfrac{18}{4} = 4.5 \)
So the sequence is:
\( 2,\ 6.5,\ 11,\ 15.5,\ 20 \)
5. Examples
These examples show how arithmetic sequences appear in both numerical patterns and everyday situations.
5.1. Simple AP Examples
- \( 4, 9, 14, 19, 24, \ldots \)
- \( 50, 45, 40, 35, 30, \ldots \)
- \( -2, 1, 4, 7, 10, \ldots \)
Each pattern follows the idea of adding the same difference.
5.2. Real-Life AP Patterns
Many real-world patterns form arithmetic sequences. Common examples include:
- Daily increase in saved money by a fixed amount
- Seats arranged in rows increasing by the same number
- Steps of a ladder placed at equal distances
These patterns grow steadily, just like arithmetic sequences.