Difference between revisions of "Recursion"

Revision as of 21:49, 21 June 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

A combinatorical use of recursion: AIME 2006I/11
Use of recursion to compute an explicit formula: AIME 2006I/13

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-'''Recursion''' is defining something in terms of a previous term, function, etc. For example, the famous [[Fibonacci sequence]] is defined recursively. If we let <math>F_n</math> be the nth term, the sequence is: <math>\displaystyle F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on. A recursive definition is: <math>F_{n+1}=F_{n}+F_{n-1}</math>. That is a symbolic way to say "The next term is the sum of the two previous terms".
+'''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 <math>F_n</math> be the <math>n</math>th Fibonacci number, the sequence is defined recursively by the relations <math>F_0 = F_1 = 1</math> and <math>F_{n+1}=F_{n}+F_{n-1}</math>.  (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: <math>\displaystyle F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on.
+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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Difference between revisions of "Recursion"

Revision as of 21:49, 21 June 2006

Examples

See also