CSE
1400
Applied Discrete Mathematics
Sequences
Department of Computer Sciences
College of Engineering
Florida Tech
Spring
2011
1
Sequences
1
1
.
1
Operations on Sequences
2
1
.
2
Useful Sequences
3
1
.
3
Growth Rates
5
1
.
4
Defined by Recurrence Equations
6
1
.
5
Defined by Functions on the Natural Numbers
8
1
.
6
Computed by Algorithms
8
1
.
7
Non-Computable Sequence
9
Abstract
1
Sequences
Sequences are ordered lists of values. Ordinal numbers: first, second,
third, fourth, . . . , specify the positions of these values. One famous
sequence is the
Fibonacci
sequence.
h
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
i
The first, second, third, fourth, and fifth
Fibonacci
numbers are
0, 1, 1, 2, and 3. The values in the
Fibonacci
sequence can be named
f
k
where the subscript
k
denotes the position of the value. In comput-
ing, it is customary to start the subscript
k
at 0 so that the first, sec-
ond, third, fourth, and fifth
Fibonacci
numbers are named
f
0
,
f
1
,
f
2
,
f
3
,
and
f
4
.
Let
~
S
=
h
s
0
,
s
1
,
s
2
,
s
3
,
s
4
, . . .
i
be a sequence. The subscripted names
s
0
,
s
1
,
s
2
,
s
3
,
s
4
, . . . , are called
terms
and they refer to values in the first, second, third, fourth,
fifth, etc., positions of the sequence. In computing theory, sequences