The difference between each pair of consecutive terms in an arithmetic sequence is constant; this constant difference is called the common difference.
An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is crucial in defining and understanding arithmetic sequences.
Key Concepts:
- Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
- Common Difference (d): The constant difference between consecutive terms in an arithmetic sequence.
Formula:
If we denote the first term of an arithmetic sequence as a1 and the common difference as d, then the nth term, an, can be expressed as:
an = a1 + ( n - 1 ) d
Example:
Consider the arithmetic sequence: 2, 5, 8, 11, 14...
Here:
- a1 = 2 (the first term)
- d = 5 - 2 = 3 (the common difference)
Each term is obtained by adding the common difference, 3, to the previous term.
Significance of the Common Difference:
- Defines the sequence: Knowing the first term and the common difference completely defines the arithmetic sequence.
- Predicts future terms: The common difference allows you to predict any term in the sequence.
- Identifies arithmetic sequences: If the difference between consecutive terms is constant, the sequence is arithmetic.
Summary:
The consistent difference between consecutive terms defines an arithmetic sequence and is known as the common difference. Understanding this concept is essential for analyzing and working with arithmetic sequences.
