The common difference in an arithmetic sequence of fractions is found by subtracting any term from the term that immediately follows it.
Here's a detailed breakdown:
Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the "common difference."
Finding the Common Difference with Fractions
When dealing with arithmetic sequences involving fractions, the process remains the same as with whole numbers. Here's how to find it:
- Identify two consecutive terms: Pick any two terms in the sequence that appear directly next to each other. For instance, in the sequence 1/2, 1, 3/2, 2,... we can choose 1 and 3/2.
- Subtract the first term from the second: Subtract the earlier term from the later one. Let's use our example sequence above. We have 3/2 - 1 = 3/2 - 2/2 = 1/2.
- Verify with another pair: To confirm that you have the common difference, repeat the subtraction with another pair of consecutive terms. For example, in the same sequence, you can choose 1/2 and 1. So, 1 - 1/2 = 2/2 - 1/2 = 1/2
- The result is the common difference: The result of the subtraction will be your common difference.
Example:
Consider the sequence: 1/2, 1, 3/2, 2, ...
- The first term is 1/2.
- The second term is 1, which can also be represented as 2/2.
- The third term is 3/2.
- The fourth term is 2, which can be represented as 4/2.
According to the reference, "And think of two as four halves. And it becomes even more clear that you just keep adding one half. So d which is called the common", the common difference is 1/2.
Calculation
Here are some calculations based on the above sequence to showcase the process:
| Consecutive Terms | Subtraction | Common Difference |
|---|---|---|
| 1 and 3/2 | 3/2 - 1 = 3/2 - 2/2 | 1/2 |
| 1/2 and 1 | 1 - 1/2 = 2/2 - 1/2 | 1/2 |
| 3/2 and 2 | 2 - 3/2 = 4/2 - 3/2 | 1/2 |
Key Takeaways
- The common difference can be a fraction, a whole number, or even a negative number.
- Always subtract the earlier term from the later term when calculating the common difference.
- Verify the common difference with multiple pairs of consecutive terms to ensure accuracy.
- As seen in the reference "And think of two as four halves. And it becomes even more clear that you just keep adding one half," the common difference can be seen as what you add to one term to get to the next.
