A number pattern with a constant ratio is a geometric sequence.
Understanding Geometric Sequences
Geometric sequences are characterized by a constant ratio between consecutive terms. This constant ratio is known as the common ratio. You obtain each term by multiplying the previous term by the common ratio.
Key Characteristics
- Common Ratio: The defining feature is the consistent multiplication factor between successive terms.
- Recursive Definition: A geometric sequence can be defined recursively as an = r * an-1, where 'r' is the common ratio.
- Explicit Formula: It can also be defined explicitly as an = a1 * rn-1, where a1 is the first term, 'r' is the common ratio, and 'n' is the term number.
Examples
Let's explore some examples to illustrate this concept:
- Sequence: 2, 4, 8, 16, 32...
- Common Ratio: 2 (each term is twice the previous term)
- Sequence: 1, 3, 9, 27, 81...
- Common Ratio: 3 (each term is three times the previous term)
- Sequence: 100, 50, 25, 12.5...
- Common Ratio: 0.5 (each term is half the previous term)
Formula Breakdown
To further clarify, let's break down the explicit formula an = a1 * rn-1:
| Component | Description |
|---|---|
| an | The nth term in the sequence |
| a1 | The first term in the sequence |
| r | The common ratio between consecutive terms |
| n | The position of the term in the sequence |
Importance
Geometric sequences are crucial in various mathematical and real-world applications, including:
- Finance: Calculating compound interest.
- Population Growth: Modeling exponential population increase or decrease.
- Physics: Describing radioactive decay.
In summary, a number pattern with a constant ratio, or a geometric sequence, is a sequence where each term is found by multiplying the previous term by a fixed number, known as the common ratio. Understanding this principle is fundamental for grasping many mathematical and scientific concepts.
