Sequences

A sequence is an ordered list of numbers arranged according to a specific rule or pattern. Each number in a sequence is called a term, typically denoted as a1, a2, a3, and so on, where the subscript indicates its position. The position of each term determines its value within the sequence.

Recursive sequences define each term using previous terms, whereas explicit sequences use a formula based on position. The Fibonacci sequence exemplifies recursion: each term equals the sum of the two preceding terms. This dependence on predecessors is the defining characteristic of recursive definitions.

Partial sums are the sums of the first few terms of a sequence, often expressed using summation notation. They help analyze how the cumulative total grows as more terms are added. For example, tracking weekly savings deposits shows how total savings accumulates over time through partial sums.

Sequences model real-world processes by representing ordered data following specific patterns. A bouncing ball's decreasing height forms a sequence where each height is a fixed fraction of the previous one. Sequences also represent population growth, financial investments, and other phenomena that change predictably over time.

A telescoping sequence is a special type where most terms cancel out when the partial sum is expanded. The nth partial sum often reduces to the difference between just the first and last terms, yielding a simplified closed-form expression. This property is especially useful for evaluating infinite series and proving convergence.

Mathematically, a sequence can be viewed as a function whose domain is the set of natural numbers, with each natural number mapped to a specific term. This functional representation is useful for defining sequences explicitly or recursively, providing a rigorous framework for analyzing sequence behavior and properties.

Understanding sequences, recursive definitions, and partial sums forms the foundation for exploring advanced topics such as infinite series, convergence, and mathematical induction. Sequences are critical in calculus, number theory, and other mathematical disciplines that build upon these fundamental concepts and principles.