A linear sequence differs from an exponential sequence primarily in how each term progresses from the previous one.
Here's a breakdown of the key differences:
Understanding the Core Mechanisms
| Feature | Linear Sequence | Exponential Sequence |
|---|---|---|
| Progression | Adds or subtracts a constant value (common difference) | Multiplies or divides by a constant value (common ratio) |
| Change | Constant addition or subtraction | Constant multiplication or division |
Detailed Explanation
- Linear Sequence:
- The core characteristic of a linear sequence is that you add or subtract the same number to get from one term to the next.
- This constant value added or subtracted is known as the 'common difference'.
- The resulting sequence forms a straight line when plotted on a graph (hence the term "linear").
- Example: 2, 5, 8, 11... (common difference = 3) or 10, 7, 4, 1... (common difference = -3)
- Exponential Sequence:
- An exponential sequence, on the other hand, is defined by multiplying or dividing by the same number to get from one term to the next.
- This constant value multiplied or divided is called the 'common ratio'.
- When plotted, the sequence will show a curve that increases (or decreases) dramatically.
- Example: 2, 4, 8, 16... (common ratio = 2) or 100, 50, 25, 12.5... (common ratio = 0.5)
Practical Insights
- Linear Sequences in Real Life: Consider a person saving the same amount of money each month. The cumulative savings form a linear sequence.
- Exponential Sequences in Real Life: Compound interest on savings or population growth often follow an exponential pattern.
Key Takeaway from the Reference
According to the reference provided: "An exponential sequence multiplies (or divides) by the same number each time as you go from term to term. A linear sequence adds (or subtracts) the same number each time." This reinforces the core distinction between the two types of sequences.
In summary, the key difference lies in whether you are adding/subtracting a constant value (linear) or multiplying/dividing by a constant value (exponential) to generate the sequence.
