The fixed number 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 remains constant. This constant difference is what we call the common difference. It is the value added (or subtracted, if negative) to each term to obtain the next term in the sequence.
Key Characteristics
- Consistent Pattern: In an arithmetic sequence, each subsequent number is generated by either adding or subtracting the same number from the preceding one.
- Common Difference: The constant value added (or subtracted) is called the common difference and it's denoted by ‘d’.
- Formula: The general form of an arithmetic sequence is: a, a + d, a + 2d, a + 3d, ... where 'a' is the first term and 'd' is the common difference.
Examples
Here are a few examples illustrating how the common difference works:
| Sequence | Common Difference | Explanation |
|---|---|---|
| 10, 20, 30, 40,... | 10 | Each term increases by adding 10 to the preceding term. |
| −4, −3, −2, −1,... | 1 | Each term increases by adding 1 to the preceding term. |
| 15, 12, 9, 6, ... | -3 | Each term decreases by subtracting 3 from the preceding term. |
Practical Insights
- Finding the Common Difference: To find the common difference, subtract any term from the term that immediately follows it. For example, in the sequence 10, 20, 30, 40, you can subtract 10 from 20 (20-10=10), 20 from 30 (30-20=10) or 30 from 40 (40-30=10).
- Identifying Arithmetic Sequence: When you are checking if a sequence is arithmetic, make sure the difference is constant between every pair of consecutive terms and not just a few.
Understanding the common difference is essential for identifying, creating, and working with arithmetic sequences.
