Yes, the common ratio in a geometric sequence can absolutely be negative.
Understanding Geometric Sequences and Common Ratios
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This common ratio is what determines the pattern of growth or decay within the sequence.
According to the provided reference on Geometric sequences, "in which each term is obtained from the preceding one by multiplying by a constant, called the common ratio and often represented by the symbol r. Note that r can be positive, negative or zero." This clearly states that the common ratio can indeed be negative, positive, or zero.
How Negative Common Ratios Affect Sequences
A negative common ratio introduces an oscillating pattern in a geometric sequence. Instead of consistently increasing or decreasing, the terms will alternate between positive and negative values.
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Example 1: Positive Common Ratio
Consider a sequence with a first term of 2 and a common ratio of 3. The sequence would be: 2, 6, 18, 54,... (each term is 3 times the previous term). All terms are positive.
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Example 2: Negative Common Ratio
Now consider a sequence with a first term of 2 and a common ratio of -3. The sequence would be: 2, -6, 18, -54,... Notice how the signs alternate between positive and negative.
- First term: 2
- Second term: 2 * (-3) = -6
- Third term: -6 * (-3) = 18
- Fourth term: 18 * (-3) = -54
Practical Insights
- Negative common ratios are essential in modeling scenarios where there is an alternating change, such as in certain physical or economic phenomena.
- They are also important for understanding how certain mathematical functions behave, particularly in situations that involve oscillations or alternating series.
Conclusion
The common ratio of a geometric sequence is a versatile concept that allows for both growth and oscillatory patterns, making it a valuable tool for mathematical modeling. It is clear that this common ratio can indeed be negative.
