Sequence

Overview

A sequence is a list of objects, numbers, or events arranged in a specific order. Each item in the sequence is called a term or element, and its position is significant. Sequences can be finite or infinite and are a fundamental concept across various disciplines.

Key Concepts

The defining characteristic of a sequence is the order of its elements. This order allows for specific relationships between terms, such as arithmetic or geometric progressions. Common notations include using subscripts to denote term positions, like $a_n$ for the nth term.

Deep Dive

Sequences can be defined by explicit formulas (e.g., $a_n = 2n$) or recursive formulas (e.g., $a_n = a_{n-1} + 2$, with $a_1=1$). Understanding convergence and divergence is crucial for infinite sequences, determining if they approach a specific limit.

Applications

Sequences are vital in:

• Mathematics: Series, calculus, number theory.
• Computer Science: Algorithms, data structures (arrays, lists), string manipulation.
• Finance: Time series analysis, compound interest.
• Biology: DNA sequences, population growth models.

Challenges & Misconceptions

A common misconception is confusing sequences with sets, where order does not matter. Another challenge is accurately predicting the behavior of complex sequences without proper analytical tools. The uniqueness of each term’s position is key.

FAQs

What is the difference between a sequence and a series?

A sequence is a list of terms, while a series is the sum of the terms in a sequence.

Can a sequence have repeating elements?

Yes, sequences can contain repeating elements, but their position still matters.