Lecture on Sequences: Finite and Infinite

Overview

• Objective: Define sequence, finite and infinite sequence.
• Activities: List next terms in a sequence, derive mathematical expression or rule by pattern searching.

Key Concepts

Types of Sequences

• Finite Sequence: A sequence with a last term; its domain is the set of positive integers with a defined end.
• Infinite Sequence: A sequence without a last term; indicated by ellipsis (...).

Terms in Sequences

• Each number is called a term, represented as (a_n).
• Examples:
  • First term: (a_1)
  • Second term: (a_2)
  • Third term: (a_3)

Patterns in Sequences

1. Alphabet Sequence: A, B, G, J - Skip two letters.
2. Odd Numbers: 1, 3, 5, 7 - Next: 9, 11, 13.
3. Perfect Squares: 1, 4, 9, 16, 25 - Next: 36, 49, 64.
4. Arithmetic Sequence: Add 10 each time - 5, 15, 25, 35, Next: 45, 55, 65.
5. Triangular Numbers: 1, 3, 6, 10 - Add consecutive integers.

Mathematical Expressions and Rules

Finding General Term

• General Formula: (a_n) for all terms.
• Example Sequences:
  • (a_1 = 1), (a_3 = 6), (a_5 = 15).
• Substitution example: Substitute (n) in the general formula to find specific terms.

Examples of Specific General Terms

• Sequence Formula: (a_n = (n-3)^n).
• First 5 Terms: -2, 1, 0, 1, 32.

Creating General Terms

• Alternating Sequences: ((-1)^n) causes sign alternation.
• Examples:
  • Cubes: (a_n = n^3).
  • Reciprocal: (a_n = \frac{1}{n}).
  • Alternating Multiples: ((-1)^n \cdot 5n).
  • Perfect Squares: (a_n = n^2).
  • Alternating sign sequence: ((-1)^n - 3n).

Conclusion

• Review how to derive general formulas for sequences based on observed patterns.
• Practice with different types and rules for sequences.