Yes, a linear sequence is the same as an arithmetic sequence.
Understanding Linear and Arithmetic Sequences
Let's break down why these two terms are interchangeable.
What is an Arithmetic Sequence?
An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference.
- Example: 2, 5, 8, 11, 14... (common difference = 3)
- Example: 10, 7, 4, 1, -2... (common difference = -3)
What is a Linear Sequence?
A linear sequence, as the name suggests, can be represented by a straight line when plotted on a graph. The x-axis represents the position in the sequence (1st term, 2nd term, etc.), and the y-axis represents the value of each term.
The reference provided clarifies the link: Arithmetic sequences are also known as linear sequences because plotting the position and the term values results in a straight line graph.
The Relationship Between the Two
The reason arithmetic sequences are also called linear sequences is because their formula has the form of a linear equation:
- General term of an Arithmetic/Linear Sequence: Tn = an + b where 'a' is the common difference and 'b' is a constant.
This format (y = mx + c in standard linear form) directly implies a straight-line relationship when visualized on a graph.
Why Are They Interchangeable?
The term "linear sequence" emphasizes the graphical representation while "arithmetic sequence" emphasizes the constant difference between terms. Both refer to the same type of numerical pattern. In short, if a sequence is arithmetic, it must be linear, and vice versa.
Summary
| Feature | Arithmetic Sequence | Linear Sequence |
|---|---|---|
| Definition | Constant difference between terms. | Graphically represented by a straight line. |
| Key Property | Common difference. | Straight line plot when graphed. |
| Interchangeable | Yes | Yes |
| Formula | Tn = an + b | Tn = an + b |
Conclusion
Therefore, a linear sequence and an arithmetic sequence are two names for the same thing; they describe a sequence with a constant difference between terms, which graphically produces a straight line.
