To determine if a sequence is arithmetic or geometric, check for a common difference or a common ratio between consecutive terms. According to the reference, if the sequence has a common difference, it is arithmetic; if it has a common ratio, it is geometric.
Here's a breakdown:
Identifying Arithmetic Sequences
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Arithmetic Sequence Definition: A sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference.
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How to find it:
- Calculate the difference between consecutive terms.
- If the difference is the same for all pairs of consecutive terms, the sequence is arithmetic.
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Example: 2, 5, 8, 11, 14...
- 5 - 2 = 3
- 8 - 5 = 3
- 11 - 8 = 3
- 14 - 11 = 3
- Since the difference is consistently 3, this is an arithmetic sequence.
Identifying Geometric Sequences
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Geometric Sequence Definition: A sequence where the ratio between any two consecutive terms is constant. This constant ratio is called the common ratio.
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How to find it:
- Calculate the ratio between consecutive terms. (Divide a term by the term before it.)
- If the ratio is the same for all pairs of consecutive terms, the sequence is geometric.
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Example: 3, 6, 12, 24, 48...
- 6 / 3 = 2
- 12 / 6 = 2
- 24 / 12 = 2
- 48 / 24 = 2
- Since the ratio is consistently 2, this is a geometric sequence.
Summary Table
| Sequence Type | Identifying Characteristic | How to Find | Example |
|---|---|---|---|
| Arithmetic | Constant common difference | Calculate the difference between consecutive terms. | 1, 4, 7, 10... (d=3) |
| Geometric | Constant common ratio | Calculate the ratio between consecutive terms (divide term by prior term). | 2, 6, 18, 54... (r=3) |
