PATTERN AND SEQUENCE!! ヽ(≧∀≦)ﾉ

Formulating Rules for Sequences

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Introduction to sequence formation rules: Sequences are essential in mathematics, as they represent an ordered list of numbers determined by a specific rule or pattern. Understanding how to derive these rules is fundamental to working with sequences.

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Study sequences and their nth term rules:

  • Sequence: 3, 6, 9, 12Rule: 3n

  • Sequence: 1, 4, 9, 16Rule: n²

  • Sequence: 2, 5, 8, 11Rule: 3n - 1

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Next three terms for sequences:

  1. Sequence: 6.5, 7, 9, 11: Next terms: 13, 15, 17

  2. Sequence: 15, 18, 21, 24 => Next terms: 27, 30, 33

  3. Sequence: 3, 8, 18, 38 => Next terms: 78, 158, 318

  4. Sequence: 7, 10, 13, 16, 19, 22, 25 => Next terms: ...

  5. Sequence: 18, 27, 36, 45, 54, 63, 72 => Next terms: ...

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Definition of Sequence: A sequence is a set of numbers arranged in a defined order, governed by specific rules that can often be expressed as mathematical functions. Each number in the sequence is termed a "term."

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Example of Sequence: 3, 5, 7, 9, ... 11This is an arithmetic sequence, where each term increases by a common difference of 2.

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Terms in a Sequence:Example Sequence: 3, 5, 7, 9...

  • 1st term: 3

  • 2nd term: 5

  • 3rd term: 7

  • 4th term: 9This reinforces the concept that each term has a specific position in the sequence.

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More Examples of Sequences:

  • 3, 6, 9, 12 - Arithmetic sequence, rule: 3n

  • 1, 4, 9, 16 - Perfect squares, rule: n²

  • 2, 5, 8, 11 - Starts at 2 and increases by 3, rule: 3n - 1

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Steps to formulate the rule:

  1. Identify the pattern of differences between consecutive terms

  2. Determine if the sequence is arithmetic, geometric, or follows another rule.

  3. Formulate the nth term rule based on the observed pattern.

Given Sequence: 3, 6, 9, 12nth Term Rule: ...?Next Three Terms: ...?

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Working out rule step-by-step:

  • Sequence: 3, 6, 9, 12

  • Method: Identify the difference between terms (3, 3, 3) leads to a multiplication rule: 3 x n for each position.

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nth Term Rule: 3nNext three terms calculation: 15, 18, 21These values confirm the rule is consistent with the original sequence.

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Confirmation of nth Term Rule:

  • For the 20th term: 3n = 3 x 20 = 60This shows that the rule accurately represents the sequence as it maintains consistency throughout.

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Formulating the nth Term Rule for Sequence:

  • Sequence: 2, 5, 8, 11

  • Pattern found: The difference between each term is 3.

  • nth Term Rule: 3n - 1

  • Next terms: 14, 17, 20

  • 50th term: 3(50) - 1 = 149

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Square numbers Sequence:

  • Sequence: 1, 4, 9, 16

  • nth Term Rule: n²

  • Next three terms: 25, 36, 49

  • 15th term: 225Square numbers illustrate how applying specific formulas reveals deeper patterns within seemingly simple sequences.

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Direction: Write rules for given sequences and supply the next three terms:

  • Sequence: 5, 9, 13, 17

  • Sequence: 3, 8, 13, 18

  • Sequence: 6, 12, 18, 24, 30

  • Sequence: 6, 13, 20, 27

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Continuation or additional examples: It is beneficial to explore varied sequences to solidify understanding of rules and patterns. Familiarity with different types of sequences enables better problem-solving across math disciplines.