Sequence and Patterns in Sequences

Introduction

• Focus: Understanding sequences, both finite and infinite.
• Goals:
  • Define sequences
  • Identify patterns in sequences
  • Derive mathematical expressions for sequences

Sequence Basics

• Sequence: An ordered list of numbers, often a function of positive integers.
• Types:
  • Finite Sequence: Has a last term.
  • Infinite Sequence: Continues indefinitely, often represented with an ellipsis (...).
• Terms in Sequence: Each number in the sequence, denoted by its position (e.g., first term, second term, etc.).
  • Example: Sequence 5, 15, 25, 35, 45 has 5 terms.

Identifying Patterns

1. Letter Sequencing:

   • Example: A, B, G, J - Skip two letters.
   • Example: M, P, S - Continue skipping two letters.

2. Odd Number Sequence:

   • Example: 1, 3, 5, 7
   • Pattern: All odd numbers, next are 9, 11, 13.

3. Perfect Squares:

   • Sequence: 1, 4, 9, 16, 25
   • Next terms: 36, 49, 64

4. Additive Pattern:

   • Sequence: 5, 15, 25, 35
   • Pattern: Add 10 to each subsequent number.

5. Consecutive Number Addition:

   • Sequence: 1, 3, 6, 10, 15, 21, 28
   • Pattern: Add consecutive numbers.

General Rules for Sequences

• Expression Derivation:
  • Use rules to find subsequent terms.
  • Example: For a term a_n, formula might be derived as a function of n.

Deriving Terms

• Example Formula: For sequence 1, 3, 6, 10, 15:
  • General Rule: a_n = (n/2)(n+1)
  • Verification through substitution for various terms.

Solving for Specific Terms

• Example Problem: Find first 5 terms using a_n = (n - 3)^n

  • Calculated terms: -2, 1, 0, 1, 32

• Example Problem: Find first four terms and 20th term using a_n = (-1)^n / (2n - 1)

  • Calculated first terms: -1, 1/3, -1/5, 1/7
  • 20th Term: 1/39

Alternate Signs Rule

• General Term: (-1)^n for alternating positive and negative terms.

Finding General Terms

• Examples:
  • Cubes: a_n = n^3
  • Fractions with consecutive denominators: a_n = 1/n
  • Alternating multiples: a_n = (-1)^n * 5n
  • Perfect squares: a_n = n^2
  • Mixed signs and arithmetic sequence: a_n = (-1)^n * 3n

Conclusion

• Action Items:
  • Practice identifying patterns in sequences.
  • Develop general terms for more complex sequences.
• Next Steps:
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