The key difference lies in how each sequence is generated: geometric sequences use multiplication or division by a common ratio, while quadratic sequences exhibit a common second difference.
Understanding Geometric Sequences
Geometric sequences are created by repeatedly multiplying (or dividing) each term by a constant value, known as the common ratio (r).
- Definition: A sequence where each term is found by multiplying the previous term by a constant.
- Common Ratio (r): The constant value used to multiply each term.
- Example: 2, 4, 8, 16, 32... (r = 2). Each term is twice the previous term.
Understanding Quadratic Sequences
Quadratic sequences have a common second difference. This means that while the differences between consecutive terms are not constant, the differences between those differences are constant.
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Definition: A sequence where the general term can be represented by a quadratic expression of the form an2 + bn + c, where a, b, and c are constants.
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Common Second Difference (d2): The constant difference between the differences of consecutive terms.
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Example: 1, 4, 9, 16, 25... (The sequence of square numbers).
- First differences: 3, 5, 7, 9...
- Second differences: 2, 2, 2... (This is the common second difference)
Key Differences Summarized
| Feature | Geometric Sequence | Quadratic Sequence |
|---|---|---|
| Generation | Multiplication/Division by common ratio | Represented by a quadratic expression; constant 2nd difference |
| Common Property | Common Ratio (r) | Common Second Difference (d2) |
| General Term | Often expressed as a*r(n-1) | Often expressed as an2 + bn + c |
| Example | 3, 6, 12, 24... (r = 2) | 2, 5, 10, 17... (d2 = 2) |
| Reference | Generated by multiplying or dividing by the same amount each time – they have a common ratio r. | Quadratic sequences have a common second difference d2. |
