RUMM - Recursion and Linear Functions Lesson
Recursion and Linear Functions
Adapted from Course materials (IV.B Student Activity Sheet 2) for AMDM developed under the leadership of the Charles A. Dana Center, in collaboration with the Texas Association of Supervisors of Mathematics and with funding from Greater Texas Foundation.
5, 10, 15, 20...
This is an example of an arithmetic sequence. How is this sequence being determined? Five is added to the previous term. 5 + 5 = 10, 10 + 5 = 15, 15 + 5 = 20 and so on.
To find the next term, we add 5 to the previous term. Arithmetic sequences have constant differences between consecutive terms. What information is required to find the value of the 10th term?
To find the value of the 10th term we need the values up to the 9th term. But, instead of counting by fives up to the 9th term, is it possible to find the 10th term without knowing the 9th term?
Yes, a function rule, such as counting by 5s, is an input-output model. It defines the nth term in terms of n, not the previous term. An explicit function rule represents a functional relationship, whereas a recursive rule is based on an iterative process.
Write a function for the sequence: 5, 10, 15, 20... Think about the constant rate of change and the starting value.
Step 1: Find the constant rate of change of the sequence.
Answer: Each term is increased by 5.
Step 2: Find the starting value
Answer: In this case, 5 - 5 = 0
Step 3: Use the slope-intercept formula (y=mx+b) to structure a function rule
Answer: m = the constant rate; b = the value for the 0th term
m=5, b=0
Step 4: Using sequence notation, n represents the term and tn represents the value of the nth term
Answer: tn=5n where n is an integer > 0
Step 5: What is the value of the 10th term?
Answer: To find that number, we substitute 10 for n
t10=5(10)=50
What are the advantages and disadvantages of a recursively defined rule? When are recursive rules more useful than function rules? When are function rules more useful?
- Advantages: It is simple to write and apply a recursive rule for a small number of terms at the beginning of a sequence.
- Disadvantages: You must know the previous term; if n is large, this is a problem.
Recursive Rules
A recursive rule defines a term in a sequence using the previous term and is based on an iterative process. A function rule defines a term in a sequence using the term number and is based on an input-output model. Both recursive rules and function rules that model linear data use the start value and the constant difference in the terms. Depending on how n is defined in a recursive rule, the start value may be the value of the zero term or the first term. The zero term is the same as the y-intercept seen in the function rule.
Recursive rules are often easier to write and are useful if the terms needed are at the beginning of the sequence. If you need to find the value of a term further along in the sequence, the recursive rule may become cumbersome. In that case, the explicit function rule is more useful. An extension is that recursive rules often yield very efficient programming algorithms.
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