Sequence Basics and Patterns

Overview

This lecture covers the basics of sequences, including definitions of finite and infinite sequences, identifying patterns, and deriving formulas (general terms) for generating sequence terms.

Introduction to Sequences

• A sequence is an ordered list of numbers, functioning as a mapping from positive integers (positions) to values.
• Two types of sequences: finite (has a last term) and infinite (continues without end, often shown with ellipsis "...").
• Each number in a sequence is called a "term," and the position is called the "n-th term."

Recognizing Patterns in Sequences

• Sequences may involve letters or numbers with specific patterns (e.g., skipping letters, increasing by a fixed value).
• Examples:
  • Letters: Skipping two letters between terms.
  • Odd numbers: 1, 3, 5, 7, ...
  • Perfect squares: 1, 4, 9, 16, 25, ...
  • Adding a fixed value: 5, 15, 25, 35 (add 10 each time).
  • Triangular numbers: Add consecutive integers to the previous term.

General Term (Formula) of a Sequence

• The general term (aₙ) is a formula expressing the n-th term in terms of n.
• Example: For a pattern, test if the proposed rule matches known terms.
• For aₙ = n(n+1)/2, substituting n = 1, 2, 3, 4, 5 gives 1, 3, 6, 10, 15.
• Given a general term, plug in n = 1, 2, ... to get sequence values.

Finding Terms from a General Term

• To get specific terms, substitute n into the general term.
• Example: For aₙ = (n-3)^n, the first 5 terms are found by setting n = 1, 2, 3, 4, 5.
• For alternating sign sequences, use (-1)ⁿ in the general term.

Deducing the General Term from a Sequence

• Recognize if terms are cubes, squares, multiples, or reciprocals.
• Examples:
  • Cubes: aₙ = n³ for 1, 8, 27, 64, 125.
  • Reciprocals: aₙ = 1/n for 1, 1/2, 1/3, ...
  • Alternating sign: aₙ = (-1)ⁿ × 5n for -5, 10, -15, 20, ...
  • Squares: aₙ = n² for 1, 4, 9, 16, 25.
  • Alternating sign with linear multiples: aₙ = (-1)ⁿ × 3n.

Key Terms & Definitions

• Sequence — An ordered list of numbers following a specific rule.
• Finite Sequence — A sequence with a last term.
• Infinite Sequence — A sequence with no last term, continuing indefinitely.
• Term — A specific number in a sequence.
• General Term (aₙ) — A formula that defines the n-th term of a sequence.
• Ellipsis (...) — Indicates the sequence continues infinitely.
• Alternating Sequence — Sequence where the sign changes for each term, often using (-1)ⁿ.

Action Items / Next Steps

• Practice finding the general term and first few sequence terms for provided patterns.
• Complete exercises on distinguishing finite and infinite sequences.
• Review and memorize formulas for common sequences (squares, cubes, triangular numbers, alternating signs).