An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the "common difference."
Here’s an example of an arithmetic sequence:
Sequence: 3, 7, 11, 15, 19, 23
Common Difference: The difference between consecutive terms is 4 (7 - 3 = 4, 11 - 7 = 4, 15 - 11 = 4, 19 - 15 = 4, 23 - 19 = 4).
Explanation of what makes it arithmetic:
- Constant Difference: In this sequence, the difference between each pair of consecutive terms is always the same (4 in this case). This is the defining characteristic of arithmetic sequences.
- Linear Relationship: If you were to graph the terms of the sequence, they would form a straight line, demonstrating a linear relationship between the terms.
- Formula: The n-th term of an arithmetic sequence can be expressed using the formula: \[ a_n = a_1 + (n - 1) \cdot d \] where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. For this sequence, \( a_1 = 3 \) and \( d = 4 \).
Thus, the given sequence (3, 7, 11, 15, 19, 23) is an arithmetic sequence with a common difference of 4.
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