To find the common difference in an arithmetic sequence, simply subtract any term from the term that immediately follows it. This consistent difference between consecutive terms is what defines an arithmetic sequence.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between any two successive members is a constant. This constant is known as the "common difference." The formula for an arithmetic sequence is given by:
an = a1 + (n - 1)d
where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
Finding the Common Difference: Step-by-Step
Here's how to find the common difference (d):
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Identify consecutive terms: Choose any two terms in the sequence that are directly next to each other. For example, if your sequence is 2, 5, 8, 11, you could pick 5 and 8, or 2 and 5, etc.
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Subtract the earlier term from the later term: As shown in the reference video, to find the common difference you subtract the earlier term from the later term. For example, using the terms 8 and 5 from our example, you'd do: 8 - 5 = 3. Or if you use 5 and 2 you would do: 5 - 2 = 3.
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Verify consistency: Ensure that the difference is consistent throughout the sequence. Check another pair of consecutive terms to confirm you get the same value.
Example: Using the Reference
According to the reference, to find the common difference in a sequence such as 97, 86..., you would subtract 86 - 97 which equals -11. Therefore, the common difference (d) of this arithmetic sequence is -11.
Practical Insights and Solutions
- Always subtract in the correct order: Subtract the term that comes before from the term that comes after in the sequence.
- Consistent Difference: The common difference must be constant throughout the sequence. If you find differing results between pairs of consecutive terms, the sequence is not arithmetic.
- Negative Difference: A negative common difference indicates a decreasing arithmetic sequence.
- Zero Difference: A zero common difference indicates a constant sequence where all the terms are the same.
Summary
| Steps | Description | Example |
|---|---|---|
| 1. Identify Consecutive Terms | Choose two terms that follow each other in the sequence. | 86 and 97 |
| 2. Subtract | Subtract the earlier term from the later term. | 86 - 97 = -11 |
| 3. Verify | Check another pair of consecutive terms to confirm the same result. |
By following these simple steps you can easily find the common difference in any arithmetic sequence.
