Discrete Mathematics CS 2610 February 24, 2009 – part 1

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.

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….}

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,…}

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

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+

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)

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

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

OEIS Online Encyclopedia of Integer Sequences http://www.research.att.com/~njas/sequences/ Or just google for “integer sequences” (try it with my sequence: 1, 2, 4, 7, 13, 26, 50, …)

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.)

Summations Example 5 3 2 å = i

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

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

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