Difference between revisions of "Recursion"

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Often, it is convenient to convert a recursive definition into a closed-form definition.  For instance, the sequence defined recursively by <math>\displaystyle a_0 = 1</math> and <math>a_n = n\cdot a_{n - 1}</math> for <math>n > 0</math> also has the closed-form definition <math>\displaystyle a_n = n!</math> (where "!" represents the [[factorial]] function).
 
Often, it is convenient to convert a recursive definition into a closed-form definition.  For instance, the sequence defined recursively by <math>\displaystyle a_0 = 1</math> and <math>a_n = n\cdot a_{n - 1}</math> for <math>n > 0</math> also has the closed-form definition <math>\displaystyle a_n = n!</math> (where "!" represents the [[factorial]] function).
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== Examples ==
 
== Examples ==
  
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* [[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]]
 
* 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]]
  
=== See also ===
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== See also ==
  
 
* [[Combinatorics]]
 
* [[Combinatorics]]
 
* [[Sequence]]
 
* [[Sequence]]
 
* [[Induction]]
 
* [[Induction]]

Revision as of 12:16, 25 July 2006

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: $\displaystyle F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8$, and so on.

Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by $\displaystyle a_0 = 1$ and $a_n = n\cdot a_{n - 1}$ for $n > 0$ also has the closed-form definition $\displaystyle a_n = n!$ (where "!" represents the factorial function).


Examples


See also