Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
ITERATION
A sequence is called iterative if the output of each step becomes the input for the next step in the
process. Thus, sequences that are defined recursively are iterative sequences. Sometimes, this is
simply a convenient way to define the sequence. Here, we'll study another aspect of iterative
sequences. We want to look at the long-range behavior of a sequence. In some cases, this is
dependent upon the value of the initial term, or seed, that is used.
1.
 1

Consider the sequence: xn  
1
1

 x
n1

a.
if n  1
if n  1
Find the exact values of the first few terms of the sequence.
x1 =
x2 =
x3 =
x4 =
x5 =
b.
On your calculator, type: 1 Enter . Then type: 1 + 1 / Ans Enter , Enter ,
etc., pressing Enter repeatedly to see the successive terms of the sequence. (Be sure to
obtain decimal approximations after some point.) What happens to the terms of the
sequence as n   (as n approaches infinity)? That is, as you look at more and more
terms, what happens to these values?
c.
Repeat the process with seeds other than 1. What happens?
Seq & Ser. 5.1
Rev. S06
Mathematical Investigations IV
Name
Consider the iterative sequence given by
 c
xn = 
 3xn 1  1
2.
if n  1
if n  1
, where c is called the seed.
Pick three values for c where c > 2 and write out several of the terms of each resulting
sequence generated by the seed. Predict what will happen as n .
a.
x1 = c =
, x2 =
x3 =
As n , x n 
b.
x1 = c =
, x2 =
.
x3 =
As n , x n 
c.
x1 = c =
, x2 =
x3 =
.
x1 = c =
, x2 =
x3 =
As n , x n 
b.
x1 = c =
, x2 =
c.
x1 = c =
, x2 =
x3 =
x4 =
.
x3 =
As n , x n 
x4 =
.
In problem 2, the elements in the sequences get larger and larger, diverging to positive
infinity. In problem 3, the elements diverge to negative infinity. Is there a seed value that
does not cause the sequence to diverge to either positive or negative infinity? If so, find
it. If not, explain why it does not exist.
Seq & Ser. 5.2

x4 =
.
As n , x n 
4.
x4 =
Now pick three values for c where c < 0 and write out several of the terms of the
sequences generated by those seeds. Predict what will happen as n .
a.

x4 =
.
As n , x n 
3.
x4 =
Rev. S06
Mathematical Investigations IV
Name
5.
6.
 c if n  1
Consider yet another sequence: xn = 
 xn 1 if n  1
Pick various seed values. (What values are possible for c?) In each case, describe what
happens as n increases without limit.
a.
x1 = c =
As n , x n 
.
b.
c=
As n , x n 
.
c.
c=
As n , x n 
.
d.
c=
As n , x n 
.
e.
c=
As n , x n 
.
f.
What do you notice? (Be sure to include the specific cases for the seeds 0 and 1.)
Repeat exercise 5 with the sequence:
 c

xn =  xn1
 2  3
if n  1
if n  1
Pick various seed values and describe what happens as n increases without limit.
a.
c=
As n , x n 
.
b.
c=
As n , x n 
.
c.
c=
As n , x n 
.
d.
What do you notice?
Seq & Ser. 5.3
Rev. S06
Mathematical Investigations IV
Name
7.
In problem 4, you might have discovered that if k =
constant sequence,
1
, then the sequence becomes the
2
1 1 1
1
, , , .... The value is called a fixed point for the sequence in
2 2 2
2
problem 3.
Fixed Points
A fixed point for a given sequence is a value k such that
if k is used as a seed,
then the sequence is constant.
In other words, when the input is equal to the output,
then that value is called a fixed point.
In problems 5 and 6, as n , each sequence converges (gets very, very close) to a
fixed value or fixed point.
What is that value for problem 6?
Problem 5 actually has two fixed points. What are they? Which seed values lead to
which fixed point?
Seq & Ser. 5.4
Rev. S06
Mathematical Investigations IV
Name
Problems 2 through 6 demonstrate that the ultimate behavior of a sequence (behavior as n   )
can vary with the choice of seed. Specifically these problems have illustrated divergence to
infinity (Problems 2 and 3), equilibrium (Problem 4 with seed of ½), and approach to a stable
fixed point (Problems 5 and 6). Surprisingly these are the only behaviors possible for a wide
variety of sequences.
Another way to explore possible sequence behavior is to use a fixed seed (e.g. x1  1 ) and vary a
parameter in the sequence definition. This parameter typically represents a quantity that could be
carefully varied in a physical experiment like the resistance of an electrical circuit, the ph of a
chemical solution, or the growth rate of a bacterial population (for example, by varying the
amount of available food). Following this final example, consider the sequence

xn = 

8.
e
 rxn1
if n  1
Pick several values of r between one and two and describe what happens as n increases
without limit. That is, determine the fixed point. Keep track of your trials in a table:
1 r  2
9.
if n  1
1
fixed point
Do you observe a relationship between r and
the fixed point? Can you explain it?
Repeat this process for values of r between three and four ( 3  r  4 ). What do you
observe? Can you explain it? How is this different than or similar to problem 8?
Clearly a change in behavior occurs for some value of r between two and three (can you find it?)
Such a change is called a bifurcation. Ask your teacher to explore behaviors and bifurcations
further in the optional sequence sheets—some very interesting things can happen…
Seq & Ser. 5.5
Rev. S06