1. Discrete Mathematics CS 2610
February 24, 2009 – part 1

2. Sequences (Section 2.4)
Def. :A sequence is a function from a subset of integers I to a set S, (I  Z)
f:IS
Usually, the domain I is either a set of positive or non- negative consecutive integers {1,2,3…} or {0,1,2,3…}.
We will usually be using as the domain of I the sequence:
I = {i  Z | i > 0}
Notation:
Let i  I, the image f(i) is denoted as ai, where ai  S
ai is called a term of the sequence
{ai} represents the entire sequence
Note:
If the domain I is finite, the sequence is finite, otherwise the sequence is infinite.

3. Sequences Examples: Let the sequence {ai} be defined as ai = i + 3:
Terms: a1, a2, a3, …
Sequence {ai}: { 4, 5, 6, 7, 8….}
ai = i2:
Sequence {ai}: { 1, 4, 9, 16, 25….}
ai = 1/i:
Sequence {ai}: { 1, 1/2, 1/3, 1/4, 1/5….}

4. Sequences Def.:An arithmetic progression is a sequence of the form
a, a + d, a + 2d, a + 3d,…
where a  R is the initial term, and d  R is the common difference,
Observe that if I = {i where i >= 0 },
ai = a + i*d
ai+1 = ai + d
Example:
Let d = 3, {an} such that a=2, d=3
{an} = {2, 5, 8, 11, 14,…}

5. Sequences Def.: A geometric progression is a sequence of the form
a, ar, ar2, ar3,…
where a  R is the initial term, and r  R is the common ratio.
Observe that if I = {i | i >=0 },
ai = ari
ai+1 = air, where a is the first term
It grows exponentially

6. Sequences Sequences (non-geometric & non-arithmetic)
Fibonacci sequence
Fi = Fi-1 + Fi-2, n > 2
where F1 = 1 and F2 = 1
Each term is the sum of the previous two terms
{Fi} : { 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … }
The pairs of rabbits on an island problem
ai = in, n is a positive integer constant
ai = i!, i  Z+

7. Fibonacci Sequence (aside)
This sequence is seen frequently in Nature
Seashells
Arrangement of seeds (pine cone)
Leaf and leaf segments
Rabbit Pairs: start with one pair, offspring appear after 2nd month (so in month 3 we have two pair, month 4 we have three pair, month five we have 5 pair, etc.)
Aka, Leonardo of Pisa, filius Bonacci (son of Bonacci)

8. Fibonacci Sequence (aside)
In the limit as n  , the ratio between successive terms of the Fibonacci sequence approaches 1.618
This is called the “golden ratio”
Ratio of human leg length to arm length
Ratio of successive layers in a conch shell

9. Some Useful Sequences n2 = 1, 4, 9, 16, 25, 36, …

10. OEIS Online Encyclopedia of Integer Sequences
Or just google for “integer sequences”
(try it with my sequence: 1, 2, 4, 7, 13, 26, 50, …)

11. Summations
Let {ai} be a sequence. We can create the following summation of this sequence
i is called the index of summation
j  Z+ is the lower bound (or limit)
k  Z+, k  j is the upper bound
(Also have ∏ for product.)

12. Summations
Example 5 3 2 å = i

13. Summations Example: The limit is defined as
Let {ai} be an infinite sequence,
The limit is defined as
The limit may or may not converge to a value
Example:
does not converge

14. Summations Let X = {x1, x2, …} and f(x) be a function on X
if X = {x | P(x)},

15. Summations Examples X = {0, 2, 4, 6, 8},
P(x): x is a positive integer divisible by 3 or 11