To find common differences, particularly in sequences or series of fractions, you generally need to determine if the sequence is arithmetic. If it is, the common difference is the constant amount added to each term to get the next. Here's how to approach it:
Identifying an Arithmetic Sequence with Fractions
First, determine if the fractional sequence is arithmetic. This means there's a constant difference between consecutive terms.
Steps to Calculate the Common Difference:
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Examine the Fractions: Look at the sequence of fractions you have. For example: 1/2, 3/4, 1, 5/4...
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Convert to a Common Denominator: If the fractions do not have the same denominator, find a common denominator and rewrite them. This makes comparisons and subtraction easier. Our example already has denominators of 2 and 4, so lets rewrite 1/2 as 2/4 making it, 2/4, 3/4, 4/4, 5/4.
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Subtract Consecutive Terms: Subtract any term from the subsequent term. According to the reference:
- "To find the common difference, simply subtract the first term from the second term, or the second from the third, or so on..."
- For example:
- Subtract the first term (2/4) from the second term (3/4): 3/4 - 2/4 = 1/4.
- Subtract the second term (3/4) from the third term (4/4): 4/4 - 3/4 = 1/4.
- Subtract the third term (4/4) from the fourth term (5/4): 5/4 - 4/4 = 1/4.
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Verify Consistency: Check that you obtain the same result each time you do the subtraction between consecutive terms. If so, that is the common difference. In our example, 1/4 is the common difference.
Example Breakdown:
| Original Sequence | Common Denominator | Subtraction | Common Difference |
|---|---|---|---|
| 1/2, 3/4, 1, 5/4 ... | 2/4, 3/4, 4/4, 5/4.. | 3/4-2/4, 4/4-3/4, 5/4-4/4 | 1/4 |
Important Note:
- Consistency is Key: If the differences between consecutive terms are not constant, then the sequence is not arithmetic, and there is no single common difference.
Practical Insights
- Simplification: Always simplify fractions after subtraction to the lowest terms.
- Negative Differences: The common difference can be a negative number, indicating a decreasing sequence.
- Zero Difference: A zero common difference indicates a constant sequence (e.g., 1/2, 1/2, 1/2…).
