No, a linear sequence cannot go both up and down.
Here's why, based on the definition of a linear sequence:
According to the provided reference, a linear sequence repeatedly increases or decreases by the same amount. This constant change (either an increase or a decrease) implies that the sequence will either consistently move upwards or consistently move downwards. A sequence that goes both up and down would not maintain this constant difference between terms, and therefore would not be considered a linear sequence.
To illustrate, let's examine what constitutes a linear sequence versus a sequence that oscillates:
Examples
| Sequence Type | Description | Example |
|---|---|---|
| Linear | The difference between consecutive terms is constant. | 2, 4, 6, 8, 10... (consistently increases by 2) 10, 8, 6, 4, 2... (consistently decreases by 2) |
| Non-Linear | The difference between consecutive terms is not constant. Can go up and down. | 1, 2, 1, 2, 1... (alternates between 1 and 2) 1, 3, 2, 4, 3... (increases then decreases) |
Key Characteristics of Linear Sequences
- Constant Difference: Each term is obtained by adding (or subtracting) the same value to the previous term.
- Monotonic Trend: They exhibit a consistently increasing or decreasing pattern.
- Formulaic Representation: Can be represented by a linear equation (e.g., an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference).
Why "Up and Down" is Non-Linear
A sequence that alternates or has changing direction violates the fundamental principle of a constant common difference. The difference between consecutive terms would vary, meaning it's no longer linear.
