Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Recursion"

(changed URLs to avoid immediate viewing of the solution, also changed title of one URL)
(→‎Examples)
 
(10 intermediate revisions by 5 users not shown)
Line 9: Line 9:
  
 
* [[Mock_AIME_2_2006-2007_Problems#Problem_8 | Mock AIME 2 2006-2007 Problem 8]] ([[number theory]])
 
* [[Mock_AIME_2_2006-2007_Problems#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]]
+
*[[1994_AIME_Problems/Problem 9|1994 AIME Problem 9]]
−
* Another combinatorical use of recursion: [[2001_AIME_I_Problems#Problem_14| 2001 AIME I Problem 14]]
+
* A combinatorial use of recursion: [[2006_AIME_I_Problems#Problem_11|2006 AIME I Problem 11]]
 +
* Another combinatorial 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| 2007 AMC 12A Problem 25]]
 
* Use of recursion to count a type of number: [[2007_AMC_12A_Problems#Problem_25| 2007 AMC 12A Problem 25]]
 
* Yet another use in combinatorics [[2008_AIME_I_Problems#Problem_11| 2008 AIME I Problem 11]]
 
* Yet another use in combinatorics [[2008_AIME_I_Problems#Problem_11| 2008 AIME I Problem 11]]
 +
* [[2015_AMC_12A_Problems#Problem_22| 2015 AMC 12A Problem 22]]
 +
* [[2019_AMC_10B_Problems#Problem_25| 2019 AMC 10B Problem 25]]
 +
* [[2004_AIME_I_Problems#Problem_15| 2004 AIME I Problem 15]]
  
 
== See also ==
 
== See also ==
Line 20: Line 24:
 
* [[Sequence]]
 
* [[Sequence]]
 
* [[Induction]]
 
* [[Induction]]
−
* [http://www.artofproblemsolving.com/Wiki/index.php/Recursion Recursion]
+
* [https://artofproblemsolving.com/wiki/index.php/Recursion Recursion]
  
 
[[Category:Combinatorics]]
 
[[Category:Combinatorics]]
 
[[Category:Definition]]
 
[[Category:Definition]]

Latest revision as of 15:03, 1 January 2024

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 $F_n$ be the $n$th Fibonacci number, the sequence is defined recursively by the relations $F_0 = F_1 = 1$ and $F_{n+1}=F_{n}+F_{n-1}$. (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: $F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8$, and so on.

Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by $a_0 = 1$ and $a_n = 2\cdot a_{n - 1}$ for $n > 0$ also has the closed-form definition $a_n = 2^n$.

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

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Recursion&oldid=209397"