Recursive
| Site: | Farwell |
| Course: | Michigan Algebra I Sept. 2012 |
| Book: | Recursive |
| Printed by: | Guest user |
| Date: | Sunday, November 24, 2024, 4:13 AM |
Description
Recursive
Recursion
Recursion is the process of choosing a starting term in a sequence of numbers and repeatedly applying the same process to each term to arrive at the next term. Recursion requires that you know the value of the term immediately before the term you are trying to find. It is often used in computer programming. It helps to find the patterns that are occurring.
The nursery rhyme "This is the House That Jack Built," is an example of recursion. Please read through it and try to put into words what is happening. Select the following link to read the Mother Goose nursery rhyme:
"This is the House That Jack Built"
An everyday example of recursion is a TV camera that is focused on a scene that includes a TV monitor displaying the image recorded by the camera. The image seen on the monitor includes an image of the monitor displaying an image of the monitor and so on - that's also recursion.
In Math
1, 3, 5, 7, 9 . . .
Step 1. What is the first term?
The first term in the sequence is 1.
Step 2. What has to be done to the first term to yield the second term?
If you add 2 to the first term, you get the second term.
Step 3. Does this pattern continue?
Formulas
A recursive sequenceis an ordered list of numbers defined by a starting value and a rule. Given a list of numbers when the same operation is applied over and over, a formula or rule can be written. A recursive formula always has two parts:
- The starting value which is called
. This means the first term in the list or the function rule for term one. - The recursion equation or formula which states what operation is being done to the previous term, called
, to get to any term known as 
.
The recursive formula for the ordered list 1,3,5,7,9, . . . would be:
Example 1
Step 1. Define the first term.
Step 2. Find the pattern to get from one term to the next.
4 + 8 = 12
12 + 8 = 20
20 + 8 = 28
28 + 8 = 36
Step 3. Write a recursive function to represent this pattern.
Step 4. Put the two parts together into one recursive formula.
The recursive formula is: 
Example 2
Step 1. Define the first term.
Step 2. Find the pattern to get from one term to the next.
12 - 3 = 9
9 - 3 = 6
6 - 3 = 3
3 - 3 = 0
Step 3. Write a recursive function to represent this pattern.
Step 4. Put the two parts together into one recursive formula.
The recursive formula is: 
Example 3
Find the recursive formula given the following table:
In this example, the initial term is the 0 term, not 1. Most of the time, recursive formulas begin with either term 0 or 1. There are other ways to show the initial term including 
for the zero term or 
for the first term. Other letters may be used as well. In this case, 
.
Now find the pattern in the table. From one term to the next, a constant amount is not added but multiplied.
In this case, to go from one term to the next, multiply by 3. Therefore, the recursive formula is:
;
Video Lesson
Recursive Functions Video
Practice
Answer Key
Sources
Amherst University, ""This is the House..."." http://www3.amherst.edu/ ~rjyanco94/literature/mothergoose/rhymes/thisisthehousethatjackbuilt.html (accessed 08/25/2010).
Holt, Rinehart, and Winston, "Lesson Tutorial." http://my.hrw.com/ math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=6496/6496.xml (accessed 08/25/2010).
KeyMath.com, "Recursive Sequences." http://www.keymath.com/x3293.xml (accessed 09/11/2010).
Regents Prep, "Recursive Sequences." http://regentsprep.org/ Regents/math/algtrig/ATP3/Recursive.htm (accessed 08/27/2010).