A number sequence is an ordered list of numbers following a specific pattern or rule. Understanding these rules is key to predicting future terms in the sequence. There are several types of sequences, and the rules governing them vary.
Common Types of Number Sequences and Their Rules
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Arithmetic Sequences: These sequences follow a rule where a constant value (called the common difference) is added or subtracted to each term to obtain the next.
- Example: 2, 5, 8, 11, 14... (Common difference = 3)
- Rule: Add 3 to the previous term.
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Geometric Sequences: These sequences follow a rule where each term is multiplied or divided by a constant value (called the common ratio) to obtain the next.
- Example: 3, 6, 12, 24, 48... (Common ratio = 2)
- Rule: Multiply the previous term by 2.
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Fibonacci Sequence: A unique sequence where each term is the sum of the two preceding terms.
- Example: 1, 1, 2, 3, 5, 8, 13...
- Rule: Add the two previous terms together.
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Other Sequences: Many other sequences exist with more complex rules, involving squares, cubes, factorials, or combinations of arithmetic and geometric progressions. These often require more detailed analysis to identify the rule.
Identifying the Rule
To determine a sequence's rule, look for patterns:
- Calculate differences or ratios between consecutive terms: If a constant difference emerges, it's likely an arithmetic sequence. A constant ratio indicates a geometric sequence.
- Look for relationships involving squares, cubes, or other powers: If the differences between terms aren't constant, check for patterns involving exponents.
- Consider recursive relationships: The term might depend on previous terms, as in the Fibonacci sequence.
Example: Finding the Rule
Let's analyze the sequence: 1, 4, 9, 16, 25...
The differences between consecutive terms are 3, 5, 7, 9... This isn't constant, but notice that the terms are perfect squares (1², 2², 3², 4², 5²...). Thus, the rule is: The nth term is n².
As noted in the provided references, number sequences are sets of numbers following a pattern or rule, with arithmetic and geometric sequences being the most common. Each number is a term, and identifying the underlying rule is crucial for understanding and extending the sequence.
