An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. Here are some examples:
Examples of Arithmetic Sequences
- Example 1: 2, 4, 6, 8, 10, 12, 14, 16, 18,... This sequence increases by a constant difference of 2.
- Example 2: 1, 5, 9, 13, 17, 21, 25, 29, 33,... Here, the constant difference is 4, as highlighted in the provided reference: "The constant value can be derived by taking the difference between any two adjacent terms".
- Example 3: 1, 8, 15, 22, 29, 36, 43, 50, ... In this sequence, each term is 7 more than the previous term.
- Example 4: 5, 15, 25, 35, 45, 55, 65, 75, ... This example has a common difference of 10.
- Example 5: 12, 24, 36, 48, 60, 72, 84, 96, ... The common difference in this sequence is 12.
Key Characteristics of Arithmetic Sequences
- Common Difference: The defining feature of an arithmetic sequence is the constant difference between consecutive terms.
- Linear Progression: Because the difference is constant, the numbers increase (or decrease) in a linear fashion.
Understanding the Constant Difference
The constant difference is crucial for identifying and working with arithmetic sequences. For instance, in the sequence 1, 5, 9, 13, ..., we can obtain the common difference by subtracting any term from the following term (e.g., 5 - 1 = 4, 9 - 5 = 4, etc.).
Table Summary
| Sequence | Common Difference |
|---|---|
| 2, 4, 6, 8, 10, ... | 2 |
| 1, 5, 9, 13, 17, ... | 4 |
| 1, 8, 15, 22, 29, ... | 7 |
| 5, 15, 25, 35, 45, ... | 10 |
| 12, 24, 36, 48, 60, ... | 12 |
The arithmetic sequences above highlight the constant progression between terms, demonstrating this basic mathematical concept.
