Linear and quadratic sequences are two different types of number patterns found in mathematics. The key difference lies in how their terms change:
Linear Sequences
A linear sequence, also known as an arithmetic sequence, has a constant first difference. This means that the same amount is added (or subtracted) each time to get from one term to the next.
- Definition: A sequence where the difference between consecutive terms is constant.
- Example: 2, 5, 8, 11, 14... (Here, the constant difference is +3)
- Formula: The general term can be represented as an + b, where a is the common difference and b is a constant.
- Practical Insight: Linear sequences model situations with steady growth or decline.
Quadratic Sequences
A quadratic sequence does not have a constant first difference, but rather a constant second difference. This means that when you find the differences between consecutive terms, those differences are not constant. However, the differences between those differences are constant.
- Definition: A sequence where the second differences between consecutive terms are constant.
- Example: 1, 4, 9, 16, 25... (The first differences are 3, 5, 7, 9..., and the second differences are 2, 2, 2...).
- Formula: The general term can be represented as an² + bn + c, where a, b, and c are constants.
- Practical Insight: Quadratic sequences often describe areas, or scenarios with accelerating growth.
Comparing Linear and Quadratic Sequences
| Feature | Linear Sequence | Quadratic Sequence |
|---|---|---|
| First Difference | Constant | Not Constant |
| Second Difference | Zero | Constant |
| General Term | an + b | an² + bn + c |
| Growth Rate | Constant | Accelerating (or decelerating) |
In Summary: According to the provided reference, "Quadratic sequences have a constant second difference. Linear sequences have a constant first difference". Linear sequences involve a steady change (constant addition or subtraction), while quadratic sequences involve a change in the rate of change (the differences between terms themselves are changing in a constant manner). Geometric sequences, unlike both linear and quadratic, share common multiplying factor rather than common difference.
