Sequence Recursion

Definition

Sequence recursion is the method of defining the terms of a sequence by specifying the first few terms (base cases) and a rule (recursive formula) that relates each subsequent term to previous ones.

Why It Matters

Recursion allows us to model complex, repetitive processes with simple, local rules. It is the foundation for everything from financial compounding algorithms to generating fractal geometry.

Core Concepts

• Recursive Formula: Defines the $n$-th term based on preceding terms (e.g., $a_n = n \cdot a_{n-1}$ for $n!$).

• How to read: “The a n equals n times a n minus 1.”
  • Meaning / when to use: Each term built from previous ones—defines factorials, Fibonacci, and iterative processes.

• Factorials: $n! = n(n-1)\dots(1)$, with the base case $0! = 1$.

• How to read: “N factorial equals n times (n-1) times … times one; zero factorial equals one.”
• Meaning: Product of all positive integers up to $n$; $0! = 1$ by convention.

• Base Case Dependency: A recursive sequence cannot be computed without its initial conditions; a missing base case makes the sequence undefined.

Connected Concepts

• Sequence Definition
• First Principles Thinking