A pattern with a constant ratio is a geometric sequence.
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number, called the constant ratio (r).
Key Characteristics
- Constant Multiplication: Unlike arithmetic sequences where you add a constant difference, in geometric sequences, you multiply by a constant ratio to get from one term to the next.
- Constant Ratio (r): The ratio between any two consecutive terms is always the same, whether it's positive or negative.
Formula for Geometric Sequence
The general formula for a geometric sequence is:
an = a1 * r(n-1)
Where:
- an is the n-th term in the sequence
- a1 is the first term
- r is the common ratio
- n is the term position
Examples of Geometric Sequences
Here are a few examples to illustrate how geometric sequences work:
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Example 1: Positive Ratio
- Sequence: 2, 4, 8, 16, 32...
- Constant Ratio (r): 2
- Explanation: Each term is obtained by multiplying the previous term by 2.
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Example 2: Negative Ratio
- Sequence: 5, -10, 20, -40, 80...
- Constant Ratio (r): -2
- Explanation: Each term is obtained by multiplying the previous term by -2.
-
Example 3: Fractional Ratio
- Sequence: 100, 50, 25, 12.5, 6.25...
- Constant Ratio (r): 0.5 or 1/2
- Explanation: Each term is obtained by multiplying the previous term by 0.5 (or dividing by 2).
Practical Insights
Geometric sequences are not just abstract mathematical concepts. They appear in various real-world scenarios, such as:
- Compound Interest: The growth of money in a savings account with compound interest follows a geometric sequence.
- Population Growth: In some simplified models, population growth can be modeled using geometric sequences.
- Radioactive Decay: The decay of radioactive substances follows a geometric pattern.
- Fractals: Geometric series are used to define fractal shapes.
Summarized Key Points
| Feature | Geometric Sequence |
|---|---|
| Definition | Sequence with constant multiplication ratio between terms |
| Constant Ratio | The ratio between consecutive terms is consistent (r) |
| Formula | an = a1 * r(n-1) |
In summary, a pattern with a constant ratio is definitively a geometric sequence, where each term is derived by multiplying the prior term by a consistent value, the constant ratio.
