Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
NOTATION AND PRACTICE
A sequence is a function whose domain is a non-empty subset of the
integers.
Although we could use any subset of the integers, in practice we usually consider
sequences where the domain consists of consecutive integers starting at 0 or 1. Recall that
instead of using the a(n) (read "a of n") function notation, we denote the nth term in the
sequence as an (read as a "sub" n). When we start with the index n = 1, each n further
denotes the number of the term; that is, an would represent the nth term. Caution: If we start
the sequence at n = 0 or n = 5, rather than at n = 1, then we must be careful when counting
terms.
A finite sequence a1 , a2 , a3 , a4 ,
, an may be denoted as:
ak  nk  1 (an ordered list of n terms).
where ak is the kth term and k takes on integer values from 1 through n. For example:
n
2

2
3
n 0
2n  18n3
and
represents the sequence 2, 3, 6, 11
represents the sequence 7, 9, 11, 13, 15, 17
Write out the terms of each of the following sequences.
5
 n  1
1.  3 
 n n1
3.
 
6
an n 1
4
2.

n 1 1 
(1)  
4 n1

 n 2 if n  1,3,5, 7,...
where an  
6  n if n  2, 4, 6,8,...
You should also recall that some sequences may be described both recursively and explicitly:
The sequence: 1, 5, 9, 13, 17, 21, 25, . . . . can be described below:
recursively by:
explicitly by:
 1
if n  1
an  4n  3, n  1, 2,3, 4,
an  
an1  4 if n  1
Seq & Ser. 2.1
Rev. S06
Mathematical Investigations IV
Name
In problems 4 – 7, write out the terms of the given sequences.
 5
5
for n  1
4. bn n 1 defined recursively by bn  
.
for
n

1
2
b

6
 n1
5.
 an  n  3
7
where an = the sum of the interior angles of a convex polygon with n sides (an
n-gon).

xt 
at  ( x, y)
2.
y

9

t


6.
at t41 where
7.
an 5n1 where an  ( x  y)n .
8.
For the sequence an  defined in problem 5, write both explicit and recursive formulas.
[Expand each term.]
Seq & Ser. 2.2
Rev. S06
Mathematical Investigations IV
Name
9.
a. Let An  n2 , n2  be the closed interval from n2 to n 2 . Graph A4 on a number


line.
b. Let Bn = length of An. Write out the terms of the sequence  Bn n 1 .
4

x  4cos(15t ), y  4sin(15t ) 
36
10. Given the sequence at t 0 where at  ( x, y)
,
(for
t
in
degrees)


describe the graph of this sequence and determine the period of this graph if the
sequence were to continue beyond t = 36.
11. a. Find the next two terms in the sequence: 1,
4
3
7
9
,
,
10
27
,
13 ,
81
,
.
b. Find a formula that explicitly generates the terms.
12. On the number line below, graph the sequence:  xk 
6
k 0
 10 if k  0

where xk  1 x
k 1 if k  0

 2
Locate and label each point xk, where k takes on the appropriate value for each term.
x
–5
0
5
10
What do you notice about the values in the sequence as k gets larger and larger?
Seq & Ser. 2.3
Rev. S06
Mathematical Investigations IV
Name
13. Let An = the number of all subsets of {1, 2, ... , n}. (Don't forget the empty set and the
set itself.) Find each of the following.
a. Write out the terms of  Ak 
5
k 1
b. Find an explicit formula for An .
14. Carefully sketch the graphs of the terms.
 f n n1 where
3


f n  ( x, y) y  cos  nx  .
Use x    ,   and y  1,1 .
y



x

15. Graph the set of all relations for which
 x, y  x 2  y 2  n2 for n  1, 2,3, 4,5 .


Describe how the graph would change if
we considered n such that 1 ≤ n ≤ 5, n .
Seq & Ser. 2.4
Rev. S06
Mathematical Investigations IV
Name
16. Given ak = [k – 2, k + 3), determine and draw each of the following on a number line.
a. a1  a2 =
b. a1  a6 =
4
c.
k 1
4
ak =
d.
k 1
ak =
The following requires careful analysis and may involve trying several cases for different
values of n.
n
e.
k 1
ak
Write the results in interval notation.
17. The matrix below models the roads between four cities, A, B, C, and D. For example:
From "A to B" there is one route using one road, "B to A" has zero routes (because it's a
one way street), and so on. M2
A B C D
B
A
represents the number of possible
A 0 1 0 1 
routes to travel between two cities, (or
M = B 0 0 1 0 
from a city back to itself), traveling


exactly two roads. M3 yields the
C 1 0 0 1 


D
number of routes that involve
C
D 1 1 0 0 
traveling exactly 3 roads, and so on.
M 
k 3
a.
List the elements in the sequence:
b.
Find the matrix that gives the number of
ways to travel between each pair of cities
using at least one but no more than three
roads. Roads may be traveled more than
once.
c.
From the matrix in part b: A trip between what two cities has the most ways to get
from one to the other?
to
has
routes
k 1
Seq & Ser. 2.5
Rev. S06