Yes, when the common difference of an arithmetic sequence is negative, the terms of the sequence are indeed decreasing.
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference. The fundamental characteristic of an arithmetic sequence, as explained by the reference, is that you simply add the common difference to any term to get the next term. This directly affects the behavior of the sequence:
- Positive Common Difference: If the common difference is positive, each subsequent term will be larger than the previous one, leading to an increasing sequence.
- Negative Common Difference: Conversely, if the common difference is negative, each subsequent term will be smaller than the previous one, making the sequence decreasing.
Example:
Let's consider an arithmetic sequence with the first term as 10 and a common difference of -2:
- Term 1: 10
- Term 2: 10 + (-2) = 8
- Term 3: 8 + (-2) = 6
- Term 4: 6 + (-2) = 4
- And so on...
As you can see, the terms are decreasing (10, 8, 6, 4...). This illustrates that a negative common difference results in a decreasing arithmetic sequence, directly supporting the statement that "if the common difference is negative, then we say that the sequence is decreasing."
Summary:
| Common Difference | Sequence Behavior |
|---|---|
| Positive | Increasing |
| Negative | Decreasing |
Therefore, the statement is true: a negative common difference in an arithmetic sequence causes the terms to decrease.
