The key difference lies in the type of numbers allowed in their respective domains, even though both have a constant rate of change.
Domains Compared: Arithmetic Sequences vs. Linear Functions
| Feature | Arithmetic Sequence | Linear Function |
|---|---|---|
| Domain | Natural numbers or a subset of natural numbers (e.g., 1, 2, 3...) | All real numbers |
| Allowed Input Values | Usually positive integers representing term numbers (1st term, 2nd term, etc.) | Any real number (positive, negative, fractions, decimals) |
| Example | The sequence 2, 4, 6, 8... would have a domain of {1, 2, 3, 4...} representing the term number. You wouldn't ask for the "2.5th" term. | The line y = 2x + 1 is defined for x = 1, x = -3, x = 0.5, x = π, and any other real number. |
| Continuity | Discrete – only defined at specific points. | Continuous – defined at every point along the real number line. |
Further Explanation
While both arithmetic sequences and linear functions exhibit a constant rate of change (also known as a common difference in arithmetic sequences and a slope in linear functions), their domains dictate how these relationships are interpreted.
- Arithmetic Sequences: Think of these as a series of distinct, ordered values. The domain represents which term in the sequence you're referring to (1st, 2nd, 3rd, etc.). Because you can't have a "halfway" term, the domain is limited to natural numbers or a subset thereof.
- Linear Functions: These functions are defined for all real numbers. This means you can plug in any value for 'x', and the function will produce a corresponding 'y' value. This creates a continuous line when graphed.
Example
Consider the arithmetic sequence defined by an = 2n + 1.
- The domain would be the set of natural numbers {1, 2, 3, ...}. We can find the 1st term (a1 = 3), the 2nd term (a2 = 5), and so on.
Now consider the linear function f(x) = 2x + 1.
- The domain is the set of all real numbers. We can find f(1) = 3, f(2) = 5, f(0.5) = 2, f(-1) = -1, and so on. The key is that any real number can be used as an input.
In conclusion, while both arithmetic sequences and linear functions share the property of a constant rate of change, their domains are fundamentally different. Arithmetic sequences are defined over natural numbers or a subset of natural numbers, while linear functions are defined over all real numbers. This difference in domain impacts the nature of the relationship being modeled: discrete for arithmetic sequences and continuous for linear functions.
