9/13/2011 Lecture 2.5 -- Sequences 1

Lecture 2.5: Sequences*

CS 250, Discrete Structures, Fall 2011

Nitesh Saxena

*Adopted from previous lectures by Zeph Grunschlag

9/13/2011 Lecture 2.5 -- Sequences 2

Course Admin Mid-Term 1 on Thursday, Sep 22

In-class (from 11am-12:15pm) Will cover everything until the lecture on Sep

15 No lecture on Sep 20

As announced previously, I will be traveling to Beijing to attend and present a paper at the Ubicomp 2012 conference

This will not affect our overall topic coverage This will also give you more time to prepare

for the exam

9/13/2011 Lecture 2.5 -- Sequences 3

Course Admin HW1 grading delayed a bit

TA/grader was sick with chicken pox Trying to finish as soon as possible HW1 solution has been released

HW2 will be posted soon Covers chapter 2 (lectures 2.*) Will be due in about a week after the mid-term Start working on it nevertheless. Will be

helpful in preparation of the mid-term

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Outline

Sequences Summation

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SequencesSequences are a way of ordering lists of objects.

Java arrays are a type of sequence of finite size. Usually, mathematical sequences are infinite.

To give an ordering to arbitrary elements, one has to start with a basic model of order. The basic model to start with is the set N = {0, 1, 2, 3, …} of natural numbers.

For finite sets, the basic model of size n is:n = {1, 2, 3, 4, …, n-1, n }

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SequencesDefinition: Given a set S, an (infinite) sequence in S is a

function N S. A finite sequence in S is a function n S.Symbolically, a sequence is represented using the subscript

notation ai . This gives a way of specifying formulaically Note: Other sets can be taken as ordering models. The book

often uses the positive numbers Z+ so counting starts at 1 instead of 0. I’ll usually assume the ordering model N.

Q: Give the first 5 terms of the sequence defined by the formula

)2πcos( iai

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Sequence ExamplesA: Plug in for i in sequence 0, 1, 2, 3, 4:

Formulas for sequences often represent patterns in the sequence.

Q: Provide a simple formula for each sequence:a) 3,6,11,18,27,38,51, … b) 0,2,8,26,80,242,728,…c) 1,1,2,3,5,8,13,21,34,…

1,0,1,0,1 43210 aaaaa

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Sequence ExamplesA: Try to find the patterns between numbers.a) 3,6,11,18,27,38,51, … a1=6=3+3, a2=11=6+5, a3=18=11+7, … and in general

ai +1 = ai +(2i +3). This is actually a good enough formula. Later we’ll learn techniques that show how to get the more explicit formula:

ai = 6 + 4(i –1) + (i –1)2

b) 0,2,8,26,80,242,728,… If you add 1 you’ll see the pattern more clearly.

ai = 3i –1c) 1,1,2,3,5,8,13,21,34,…This is the famous Fibonacci sequence given by

ai +1 = ai + ai-1

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Bit StringsBit strings are finite sequences of 0’s and 1’s.

Often there is enough pattern in the bit-string to describe its bits by a formula.

EG: The bit-string 1111111 is described by the formula ai =1, where we think of the string of being represented by the finite sequence a1a2a3a4a5a6a7

Q: What sequence is defined by a1 =1, a2 =1 ai+2 = ai ai+1

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Bit StringsA: a0 =1, a1 =1 ai+2 = ai ai+1:

1,1,0,1,1,0,1,1,0,1,…

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SummationsThe symbol “S” takes a sequence of numbers and

turns it into a sum.Symbolically:

This is read as “the sum from i =0 to i =n of ai”Note how “S” converts commas into plus signs.One can also take sums over a set of numbers:

n

n

ii aaaaa

...2100

Sx

x2

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SummationsEG: Consider the identity sequence

ai = iOr listing elements: 0, 1, 2, 3, 4, 5,…The sum of the first n numbers is given

by:na

n

ii

...3211

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Summation Formulas – Arithmetic There is an explicit formula for the previous:

Intuitive reason: The smallest term is 1, the biggest term is n so the avg. term is (n+1)/2. There are n terms. To obtain the formula simply multiply the average by the number of terms.

2)1(

1

nnin

i

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Summation Formulas – Geometric Geometric sequences are number

sequences with a fixed constant of proportionality r between consecutive terms. For example:

2, 6, 18, 54, 162, …Q: What is r in this case?

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Summation Formulas2, 6, 18, 54, 162, …A: r = 3.In general, the terms of a geometric sequence

have the form ai = a r i

where a is the 1st term when i starts at 0.A geometric sum is a sum of a portion of a

geometric sequence and has the following explicit formula:

1)1(...

12

0

r

raarararaarn

nn

i

i

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Summation ExamplesIf you are curious about how one could prove such

formulas, your curiosity will soon be “satisfied” as you will become adept at proving such formulas a few lectures from now!

Q: Use the previous formulas to evaluate each of the following

1.

2.

103

20

)3(5i

i

13

0

2i

i

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Summation ExamplesA:1. Use the arithmetic sum formula and

additivity of summation:

103

20

103

20

103

20

103

20

355)3(5)3(5i iii

iii

2457084352

)20103(845

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Summation ExamplesA:2. Apply the geometric sum formula

directly by setting a = 1 and r = 2:

163831212122 14

1413

0

i

i

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Composite Summation

For example:

What’s

n

jji

n

i

ba11

3

1

2

1 ji

ij

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Today’s Reading Rosen 2.4