The common difference is the constant amount separating consecutive terms in an arithmetic sequence (also known as an arithmetic progression). It's found by subtracting any term from the term that follows it.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between consecutive terms remains consistent. This consistent difference is what we call the common difference.
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Example: In the sequence 2, 5, 8, 11, 14..., the common difference is 3 (5-2 = 3, 8-5 = 3, and so on).
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Finding the Common Difference: To calculate the common difference, simply subtract any term from the term immediately following it. The result will always be the same in a true arithmetic sequence. This is confirmed by multiple sources, such as Cuemath (https://www.cuemath.com/algebra/common-difference/), which states that the common difference is "the difference between any of its terms and its previous term." Similarly, Varsity Tutors (https://www.varsitytutors.com/hotmath/hotmath_help/topics/common-difference) defines it as "the constant difference between consecutive terms of an arithmetic sequence".
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Not all sequences have a common difference: It's crucial to remember that not every sequence of numbers possesses a common difference. For instance, the sequence 1, 2, 4, 8, 16... is a geometric progression, characterized by a common ratio (in this case, 2), not a common difference (https://socratic.org/questions/what-is-the-common-difference-of-1-2-4-8-16).
Practical Applications
The concept of common difference is fundamental in various mathematical applications, including:
- Predicting future terms: Knowing the common difference allows us to easily predict subsequent terms in an arithmetic sequence.
- Solving problems involving linear relationships: Arithmetic sequences often model linear growth or decay, making the common difference a key parameter in these scenarios.
