Difference between revisions of "Recursion"
(→Examples) |
|||
| Line 4: | Line 4: | ||
Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by <math>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>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>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>a_n = n!</math> (where "!" represents the [[factorial]] function). | ||
| + | |||
| + | 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. | ||
Revision as of 14:39, 24 January 2008
| 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).
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
- 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
