Problem 70 - College Algebra

Question

What is a recursion formula?

Step-by-Step Solution

Answer

A recursion formula is a formula that expresses the elements of a sequence in terms of one or more of the previous elements. It allows for the definition of complex sequences without the need to express each element individually.

Step 1: Defining Recursion

A recursion formula, also known as a recursive formula, refers to a formula that expresses the elements of a sequence or series in terms of one or more of the previous elements. In other words, it derives subsequent elements from the preceding ones.

Step 2: Illustrating Recursion With An Example

For instance, consider the sequence \(a_n\), where every element is defined by the formula: \(a_{n} = a_{n-1} + 3\), for \(n > 1\), and \(a_{1}= 1\). This is a recursion formula. Here, each term, \(a_n\), is defined using the previous term, \(a_{n-1}\).

Step 3: Usefulness of Recursion Formulas

Recursion formulas offer an efficient and effective way to define complex sequences without the need to express each element individually. They form the mathematical basis for iterative processes, and are prevalent in several areas of mathematics and computer science, such as algorithms and fractal geometry.