A sequence is not arithmetic if it does not have a common difference between consecutive terms.
Here's how to determine if a sequence is not arithmetic, drawing from the provided reference:
To establish whether a sequence is arithmetic, geometric, or neither, you must test the terms of the sequence. You specifically look for a common difference, or a common ratio, in the sequence. If neither of these tests is successful then the sequence is neither geometric nor arithmetic, it is a sequence of neither.
Steps to Determine a Non-Arithmetic Sequence
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Understand Arithmetic Sequences: An arithmetic sequence has a constant difference between any two successive terms. This constant difference is referred to as the 'common difference'.
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Calculate Differences: Take each consecutive pair of terms in the given sequence and calculate the difference between them. Subtract the previous term from the next term (i.e., term2 - term1, term3 - term2, term4 - term3, and so on).
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Test for Common Difference: If all the calculated differences are identical, then the sequence is arithmetic. However, if you find even one pair of consecutive terms with a different difference, the sequence is not arithmetic.
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Conclusion: As detailed in the reference material, if neither a common difference nor a common ratio can be found within a sequence then that sequence is classified as being neither arithmetic nor geometric.
Examples
Here are some examples to illustrate:
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Example 1 (Arithmetic): The sequence 2, 4, 6, 8, 10... is arithmetic. The common difference is 2 (4-2 = 2, 6-4 = 2, 8-6 = 2).
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Example 2 (Not Arithmetic): Consider the sequence 1, 3, 6, 10, 15...
- 3 - 1 = 2
- 6 - 3 = 3
- 10 - 6 = 4
The differences are not the same (2, 3, and 4) so this sequence is not arithmetic.
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Example 3 (Not Arithmetic): The sequence 2, 4, 8, 16... has differences of 2, 4, 8... so it is not arithmetic either.
Summary
| Sequence Type | Definition | Common Difference | How to Test |
|---|---|---|---|
| Arithmetic | Constant difference between consecutive terms | Yes | Subtract consecutive terms. If the result is always the same, then arithmetic. |
| Non-Arithmetic | Not a constant difference | No | Subtract consecutive terms. If results vary, then not arithmetic. |
By following these steps, you can confidently determine whether a sequence is arithmetic or not.
