The difference between consecutive terms in an arithmetic sequence is a constant, known as the common difference. This constant value is what defines an arithmetic sequence.
Understanding the Common Difference
As outlined in the provided reference, an arithmetic sequence is characterized by a consistent difference between any two terms that follow each other. This consistent difference is the common difference, and it's what makes the sequence "arithmetic".
Key Concepts
- Arithmetic Sequence Definition: A sequence where the difference between any two consecutive terms is constant.
- Common Difference: The constant value added to any term to generate the subsequent term.
Illustrative Examples
Let's consider a few examples to illustrate the concept:
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Example 1: In the sequence 2, 5, 8, 11, 14, ... the common difference is 3.
- 5 - 2 = 3
- 8 - 5 = 3
- 11 - 8 = 3
- And so on...
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Example 2: In the sequence 20, 15, 10, 5, 0, ... the common difference is -5.
- 15 - 20 = -5
- 10 - 15 = -5
- 5 - 10 = -5
- And so on...
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Example 3: A simple sequence like 1, 2, 3, 4, 5 ... has a common difference of 1.
- 2 - 1 = 1
- 3 - 2 = 1
- 4 - 3 = 1
- And so on...
Practical Insights
- Finding the Common Difference: To find the common difference, simply subtract any term from its immediate successor.
- Generating the Sequence: Once you know the first term and the common difference, you can generate the entire sequence by repeatedly adding the common difference.
- Linear Relationship: Arithmetic sequences represent a linear relationship, which when graphed, will form a straight line.
Key Takeaway
The common difference is the defining feature of an arithmetic sequence. It's the constant value that is added to each term to obtain the next term in the sequence. Understanding the common difference allows you to predict and work with any arithmetic sequence.
