How does a recursive algorithm operate?

A recursive algorithm operates by calling itself within its definition, allowing it to solve larger problems by breaking them down into smaller, more manageable subproblems. This self-referential approach enables the algorithm to handle tasks in a structured, often elegant manner.

For example, consider a factorial function which calculates the product of integers from 1 to n. A recursive implementation would multiply n by the factorial of (n-1), effectively reducing the problem size each time the function calls itself until it reaches a base case, such as factorial(1) which simply returns 1. This mechanism of self-calling is what distinguishes recursive algorithms from iterative ones, which rely on loops.

The other options focus on different mechanisms that aren't characteristic of recursive algorithms. While loops are a feature of iterative approaches, caching results refers more to dynamic programming than recursion directly, and dividing input into equal parts describes approaches like divide-and-conquer rather than the recursive nature itself. Thus, the essence of recursion lies in this ability to call itself, facilitating a divide-and-conquer strategy or simply reducing the problem size directly within its structure.