To find the common difference in an arithmetic sequence, you don't necessarily need a dedicated "calculator". The method involves simple subtraction, which can be done with any basic calculator or even manually.
Here's how to determine the common difference:
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Subtract any two adjacent terms. The common difference is the value obtained when you subtract a term from its subsequent term. This can be represented as an+1 - an.
- For example, according to the provided reference, you can calculate the common difference by subtracting any two consecutive terms like a2 - a1, a7 - a6, or a100 - a99.
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Verify the difference. To confirm that the sequence is indeed arithmetic, calculate the difference between multiple pairs of adjacent terms. If the result is consistent across all pairs, then that value is the common difference. If the difference isn't consistent, then the sequence isn't arithmetic.
Here's a table summarizing the process:
| Step | Description | Example |
|---|---|---|
| 1 | Identify any two adjacent terms in the arithmetic sequence. | Sequence: 2, 4, 6, 8,... Choose 4 and 2. |
| 2 | Subtract the first term from the second term (subsequent term minus the preceding term). | 4 - 2 = 2 |
| 3 | Repeat step 2 with different adjacent terms to ensure consistency. | 6 - 4 = 2, 8 - 6 = 2 |
| 4 | If the difference is the same for all pairs of adjacent terms, that value is the common difference. Otherwise, it's not an arithmetic sequence. | In the example above, the common difference is 2. |
Example:
Let's say you have the arithmetic sequence: 5, 10, 15, 20, ...
- Choose two adjacent terms: 10 and 5.
- Subtract: 10 - 5 = 5.
- Verify: 15 - 10 = 5, 20 - 15 = 5.
Since the difference is consistently 5, the common difference for this arithmetic sequence is 5.
