A geometric sequence is characterized by a constant ratio between consecutive terms. Here's a breakdown of its key characteristics:
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Constant Ratio (Common Ratio): This is the defining characteristic. Each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio, often denoted as 'r'.
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Multiplicative Relationship: Unlike arithmetic sequences (which involve addition or subtraction), geometric sequences are based on multiplication. To move from one term to the next, you multiply by the common ratio.
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General Formula: The nth term (an) of a geometric sequence can be expressed using the formula:
an = a1 * r(n-1)
where:
- a1 is the first term
- r is the common ratio
- n is the term number
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Example: Consider the sequence 2, 6, 18, 54... Here, the first term (a1) is 2, and the common ratio (r) is 3 (since 6/2 = 3, 18/6 = 3, and so on). Therefore, the 5th term would be 2 3(5-1) = 2 34 = 2 * 81 = 162.
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Increasing or Decreasing: A geometric sequence can be increasing (if r > 1) or decreasing (if 0 < r < 1). If r is negative, the sequence will alternate between positive and negative values.
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Zero: If any term in a geometric sequence is zero and r is not zero, then all subsequent terms will also be zero. However, geometric sequences typically exclude 0 as a term, because then the common ratio cannot be accurately calculated.
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Recursive Definition: A geometric sequence can also be defined recursively:
- a1 = (First Term)
- an = r * a(n-1) for n > 1
In summary, geometric sequences are multiplicative sequences where each term is obtained by multiplying the previous term by a constant common ratio. This results in a distinct pattern defined by exponential growth or decay, depending on the value of the common ratio.
