Difference between revisions of "Recursion"
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* [[Mock_AIME_2_2006-2007/Problem_8 | Mock AIME 2 2006-2007 Problem 8]] ([[number theory]]) | * [[Mock_AIME_2_2006-2007/Problem_8 | Mock AIME 2 2006-2007 Problem 8]] ([[number theory]]) | ||
* A combinatorical use of recursion: [[2006_AIME_I_Problems#Problem_11|2006 AIME I Problem 11]] | * A combinatorical use of recursion: [[2006_AIME_I_Problems#Problem_11|2006 AIME I Problem 11]] | ||
| + | * Another combinatorical use of recursion: [[2001 AIME I Problems/Problem 14| 2001 AIME I Problem 14]] | ||
* Use of recursion to compute an explicit formula: [[2006_AIME_I_Problems#Problem_13| 2006 AIME I Problem 13]] | * Use of recursion to compute an explicit formula: [[2006_AIME_I_Problems#Problem_13| 2006 AIME I Problem 13]] | ||
* Use of recursion to count a type of number: [[2007 AMC 12A Problems/Problem 25]] | * Use of recursion to count a type of number: [[2007 AMC 12A Problems/Problem 25]] | ||
Revision as of 20:44, 19 June 2008
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 
.
In computer science, recursion also refers to the technique of having a function repeatedly call itself. The concept is very similar to recursively defined mathematical functions, but can also be used to simplify the implementation of a variety of other computing tasks.
Examples
- Mock AIME 2 2006-2007 Problem 8 (number theory)
- A combinatorical use of recursion: 2006 AIME I Problem 11
- Another combinatorical use of recursion: 2001 AIME I Problem 14
- 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
