Recursion
| This is an AoPSWiki Word of the Week for Jan 17-23 |
Recursion is a method of defining something (usually a sequence or function) in terms of previously defined values. The most famous example of a recursive definition is that of the Fibonacci sequence. If we let 
be the 
th Fibonacci number, the sequence is defined recursively by the relations 
and 
. (That is, each term is the sum of the previous two terms.) Then we can easily calculate early values of the sequence in terms of previous values: 
, and so on.
Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by 
and 
for 
also has the closed-form definition 
(where "!" represents the factorial function).
Examples
- Mock AIME 2 2006-2007 Problem 8 (number theory)
- A combinatorical use of recursion: 2006 AIME I Problem 11
- Use of recursion to compute an explicit formula: 2006 AIME I Problem 13
- Use of recursion to count a type of number: 2007 AMC 12A Problems/Problem 25
