The primary difference between a geometric sequence and an arithmetic sequence lies in how each term is generated from the previous one: an arithmetic sequence adds a constant difference, while a geometric sequence multiplies by a constant ratio.
Arithmetic Sequences: Constant Difference
An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference (d).
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Formula: an = a1 + (n-1) * d
- an represents the nth term in the sequence.
- a1 represents the first term.
- d represents the common difference.
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Example: 2, 5, 8, 11, 14... (common difference = 3)
Geometric Sequences: Constant Ratio
A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio (r).
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Formula: an = a1 * r(n-1)
- an represents the nth term in the sequence.
- a1 represents the first term.
- r represents the common ratio.
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Example: 3, 6, 12, 24, 48... (common ratio = 2)
Key Differences Summarized
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Definition | Constant addition between consecutive terms | Constant multiplication between consecutive terms |
| Operation | Addition/Subtraction | Multiplication/Division |
| Constant | Common Difference (d) | Common Ratio (r) |
In simpler terms, arithmetic sequences grow (or shrink) linearly, whereas geometric sequences grow (or shrink) exponentially. The reference correctly points out the similarity between arithmetic sequences and linear functions. Geometric sequences, on the other hand, exhibit exponential growth or decay.
