Class Notes: Finding the Next Numbers in a Sequence

Objective

Learn how to identify patterns in number sequences to predict the next numbers.

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Key Patterns and Examples

1. Arithmetic Sequences

• Definition: Numbers increase or decrease by a constant amount (common difference).
• Example:
  • Sequence: 5, 8, 11, 14
  • Pattern: Add 3 each time.
  • Next three numbers: 17, 20, 23

2. Geometric Sequences

• Definition: Numbers are multiplied or divided by a constant factor (common ratio).
• Example:
  • Sequence: 3, 6, 12, 24
  • Pattern: Multiply by 2 each time.
  • Next three numbers: 48, 96, 192

3. Arithmetic Sequence with Negative Difference

• Example:
  • Sequence: 19, 15, 11, 7
  • Pattern: Subtract 4 each time.
  • Next three numbers: 3, -1, -5

4. Geometric Sequence with Division

• Example:
  • Sequence: 320, 160, 80, 40
  • Pattern: Divide by 2 each time (or multiply by ½).
  • Next three numbers: 20, 10, 5

5. Increasing Addition Pattern (Not Pure Arithmetic or Geometric)

• Example:
  • Sequence: 1, 3, 6, 10
  • Addition increases by one each time: +2, +3, +4, etc.
  • Next three numbers: 15, 21, 28 (add 5, 6, 7 respectively)

6. Alternating Pattern

• Example:
  • Sequence: 2, 6, 5, 9, 8, 12, 11
  • Pattern alternates: +4, -1, +4, -1, ...
  • Next three numbers: 15 (+4), 14 (-1), 18 (+4)

7. Pattern Based on Adding Perfect Squares

• Example:
  • Sequence: 1, 2, 6, 15, 31
  • Pattern: Each term is the previous term plus consecutive perfect squares:
    • +1 (1²), +4 (2²), +9 (3²), +16 (4²), etc.
  • Next three numbers:
    • 31 + 25 (5²) = 56
    • 56 + 36 (6²) = 92
    • 92 + 49 (7²) = 141

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How to Identify Patterns

1. Look for constant addition or subtraction (arithmetic).
2. Look for constant multiplication or division (geometric).
3. Examine changes in adding or subtracting values (could involve squares, cubes, or an increasing pattern).
4. Check for alternating or repeating patterns.
5. Write out differences or ratios between terms to spot the trend.

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Summary

• Identify if the pattern involves addition, subtraction, multiplication, division, or more complex operations (like alternating or increasing the added value).
• Once the rule is found, use it to predict subsequent numbers in the sequence.

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For more examples and practice problems, check the provided link or the instructor’s website for additional resources on patterns, algebra, geometry, trigonometry, chemistry, or physics.