The rule or equation that determines a term in a sequence depends entirely on the type of sequence. There are several common types, each with its own defining equation.
Common Sequence Types and Their Equations
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Arithmetic Sequence: An arithmetic sequence has a constant difference between consecutive terms. The nth term (an) is given by the formula:
an = a1 + (n - 1)d
where:
- a1 is the first term
- n is the term number
- d is the common difference
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Geometric Sequence: A geometric sequence has a constant ratio between consecutive terms. The nth term (an) is given by:
*an = a1 r(n-1)**
where:
- a1 is the first term
- n is the term number
- r is the common ratio
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Fibonacci Sequence: This sequence is defined recursively, meaning each term is the sum of the two preceding terms:
an+2 = an+1 + an
The sequence begins with 0 and 1, or 1 and 1, depending on the convention used.
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Quadratic Sequences: These sequences have a second difference that is constant. Finding the equation for the nth term often involves analyzing the first and second differences to determine a quadratic expression. There's no single, simple formula like the arithmetic or geometric sequences.
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Recursive Sequences: Many sequences are defined recursively. A recursive formula defines a term based on the previous term(s). For example, a recursive formula could be expressed as:
an = f(an-1, an-2, ...)
Determining the Rule for an Unknown Sequence
To find the rule for an unknown sequence, you need to analyze the pattern of the numbers. Look for:
- Constant differences: If the differences between consecutive terms are constant, it's an arithmetic sequence.
- Constant ratios: If the ratios between consecutive terms are constant, it's a geometric sequence.
- Recursive relationships: Check if each term is related to the preceding terms through a specific operation (addition, subtraction, multiplication, etc.).
- Higher-order differences: If neither constant differences nor ratios are present, examine higher-order differences (differences of differences, differences of differences of differences, etc.). This can reveal quadratic or higher-degree polynomial relationships.
Example: The sequence 2, 5, 8, 11, ... has a constant difference of 3. Therefore, it's an arithmetic sequence, and the formula for the nth term is an = 2 + (n-1)3 = 3n - 1.
Remember that not all sequences can be described by a simple formula. Some sequences might be defined only by a recursive relationship or might not exhibit a clear pattern.
