Download Sequences Summations

Sequences and Summations
MSU/CSE 260 Fall 2009
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Sequences
Definition: A sequence is a function whose domain
D is a subset of N ={0, 1, 2, …} of the form:
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D = { k, k+1, k+2, … }, for some k  N, or
D = { k, k+1, k+2, …, k + m }, for some k, m  N
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Typically, k  { 0, 1 }.
Terminology:
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In the first case, the sequence is said to be infinite; in the
second, it is said to be finite, and of length m.
an, is called a term of the sequence and denotes the image
of the integer n.
{an} denotes the sequence.
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Do not confuse the above notation with set notation. This is a
“misuse” of the set notation.
MSU/CSE 260 Fall 2009
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A function that generates pseudo random
integers.
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Choose some prime P, say 11
Choose some starting value S, say 5.
Choose a “generator”G, say 7.
F(1) = 5 (= S)
F(k) = F(k-1) * 7 modulo 11 (F(k-1)*G mod P))
Worksheet: Run the sequence for 11, 7, 5
MSU/CSE 260 Fall 2009
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What are random numbers for?
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Assigning probabilities for events in games or
simulations.
Generating unbiased test data.
Creating novel graphics, etc.
Typical random number generator generates
random bits or maps integers into the “reals” in
the interval [0, 1). (See MATLAB or C rand.)
MSU/CSE 260 Fall 2009
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Example
1
Consider sequence {an }, where an  .
n
1
1
1
Few terms are: a1  1, a2  , a3  , a4  ,
2
3
4
MSU/CSE 260 Fall 2009
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Example
Consider the sequence { f n }, where
n
1 1 5 
1 1 5 
fn 

 


5 2 
5 2 
Find terms: f 0 , f1 , f 2 , f 3 , f 4 , f 5 , f 6 ,
MSU/CSE 260 Fall 2009
n
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Geometric Progression
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A geometric progression is a sequence of the
form:
a, ar, ar2, …, arn
where the initial term, a, and common ratio, r, are
real numbers.

A geometric progression is a discrete analogous of
the exponential function f (x) = arx.
MSU/CSE 260 Fall 2009
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Examples

The sequence {bn}, where bn= (-1)n for
n = 1, 2, 3… starts with: -1, 1, -1, 1, …
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The sequence {cn}, where cn= 2n for n = 0, 1, 2,
3… starts with: 1, 2, 4, 8, 16, … (Consider the
number of nodes in a perfect binary tree.)
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The sequence {dn}, where dn= 6·(1/3)n for
n = 1, 2, 3… starts with: 2, 2/3, 2/9, 2/27, …
MSU/CSE 260 Fall 2009
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Arithmetic Progression
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An arithmetic progression is a sequence of the
form
a, a+d, a+2d, …, a+nd
where the initial term, a, and the common
difference, d, are real numbers.

A arithmetic progression is a discrete analogous of
the linear function f (x) = dx + a.
MSU/CSE 260 Fall 2009
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Examples
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The sequence {sn}, where sn= – 1 + 4n for
n =1,2,3… starts with: 3, 7, 11, …
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The sequence {tn}, where tn= 7 – 3n for n =1, 2,
3… starts with: 4, 1, -2, …
MSU/CSE 260 Fall 2009
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Strings
Strings are sequences of the form a1a2…an,
where the codomain is the set of characters
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The length of the string is the number of its terms
The empty string is the string that has no terms
MSU/CSE 260 Fall 2009
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Special Integer Sequences
To deduce a possible formula/rule for the terms
of a sequence from initial terms, ask the
following:
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Are there runs of the same value?
Are terms obtained from previous terms by adding the
same amount or an amount that depends on the
position in the sequence?
Are terms obtained from previous terms by
multiplying by a particular amount?
Are terms obtained by combining previous terms in a
certain way?
Are there cycles among the terms?
MSU/CSE 260 Fall 2009
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Examples
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Consider the sequence
5, 11, 17, 23, 29, 35, 41, 47, 53, 59…
Describe {an}
6
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6
6
{an} = 5 +6n
n = 0, 1, 2,…
MSU/CSE 260 Fall 2009
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Examples
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Consider the sequence
1, 7, 25, 79, 241, 727, 2185, 6559, 19681,
Describe {an}
6 18 54
Answer:
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An = 3n - 2
MSU/CSE 260 Fall 2009
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Useful Sequences
nth Term
First 10 Terms
n2
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
n3
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …
n4
1, 16, 81,256,625,1296,2401,4096,6561,10000,…
2n
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
n!
1,2,6,24,120,720,5040,40320,362880,3628800,…
MSU/CSE 260 Fall 2009
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Summations
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Consider the sequence {ak}.
We define the following summation:
n
a
j m
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j
 am  am1  am2 
 an
Terminology:
j is called the index of summation,
m is the lower limit,
n is the upper limit.
MSU/CSE 260 Fall 2009
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Summations…
Note that the following are all the same:
n
n
n
a  a   a
j m
j
i m
i
k m
k


m  j n
MSU/CSE 260 Fall 2009
aj
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Worksheet problem: P 103 *35
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Given two sequences (sets?) of positive reals
{Xj} and {Yj}
Define the function
F(k) = j=1, 2, … k(Xj * Yj)
Show that this function is maximized when
both sets are sorted into ascending order. (Note
here that we are changing the order in the
sequences.)
Consider the “student credit default swap”
problem/solution!
MSU/CSE 260 Fall 2009
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Summations…
1. Distributive Law:
c  a
kA
k
 c   ak
k A
2. Associative Law:

 

(ak  bk )    ak     bk 

kA
 k A
  k A 
3. Commutative Law:

kA
ak 


( k )A
a ( k ) where  ( k ) is
any permutation of the set of natural numbers. Example:
 (a  b  k )  
0 k  n
( a  b  ( n  k ))
0 n  k  n
MSU/CSE 260 Fall 2009
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