The difference between two consecutive numbers in an arithmetic sequence is called the common difference.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is the defining characteristic of such sequences and makes them predictable and easy to work with.
Key Aspects of Arithmetic Sequences
- Constant Difference: The core principle is that each term is generated by adding the same value to the previous term.
- Predictable Pattern: Due to the constant difference, the sequence follows a linear progression, making it easy to identify terms within the sequence and even predict future terms.
Common Difference in Detail
As stated in the provided reference, the common difference is "the amount between each number in an arithmetic sequence." It's called "common" because this difference remains the same throughout the entire sequence. In other words, whether you subtract the first term from the second, the second from the third, or any other consecutive pair, you will always find the same value. This makes it the central element in analyzing and working with arithmetic sequences.
Calculation of Common Difference
To find the common difference, simply subtract any term in the sequence from the term that immediately follows it. For example:
- If an arithmetic sequence is 2, 5, 8, 11, ..., the common difference is 5 - 2 = 3, or 8 - 5 = 3, and so on.
- If an arithmetic sequence is 10, 7, 4, 1, ..., the common difference is 7 - 10 = -3, or 4 - 7 = -3, and so on.
Significance of Common Difference
The common difference is crucial because:
- Identifies the Sequence: It is the characteristic feature that defines an arithmetic sequence.
- Predicts Terms: Knowing the common difference allows us to easily predict any subsequent term in the sequence.
- Calculates the n-th Term: Formulas for finding the n-th term of an arithmetic sequence are directly based on the common difference and initial term.
Example
Let's consider the arithmetic sequence: 3, 7, 11, 15, 19...
- The difference between 7 and 3 is 4 (7 - 3 = 4)
- The difference between 11 and 7 is 4 (11 - 7 = 4)
- The difference between 15 and 11 is 4 (15 - 11 = 4)
In this sequence, the common difference is 4.
Summary
In an arithmetic sequence, the difference between any two consecutive terms is the common difference. This value remains constant throughout the sequence, making it essential for understanding and working with arithmetic progressions.
