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![Page 1: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/1.jpg)
1,
Mustafa Jarrar
Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
mjarrar©2015
Sequences & Mathematical Induction
5.1 Sequences
5.2 Mathematical Induction I
5.3 Mathematical Induction II
![Page 2: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/2.jpg)
2,
Watch this lecture and download the slides
http://jarrar-courses.blogspot.com/2014/03/discrete-mathematics-course.html
Acknowledgement: This lecture is based on (but not limited to) to chapter 5 in “Discrete Mathematics with Applications by Susanna S. Epp (3rd Edition)”.
More Online Courses at: http://www.jarrar.info
![Page 3: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/3.jpg)
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5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products Part 6: Sequence in Computer Loops and Dummy Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 4: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/4.jpg)
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كعلم الرياضيات علم الى النظر يمكن هلوتعميم الحياة في انماط اكتشاف
وقوانين؟ كنظريات االنماط !!هذهالرياضيات؟ وعلم الفن بين المشترك هو !!ما
A mathematician, like a painteror poet, is a maker of patterns. -G. H. Hardy, A Mathematicians Apology, 1940
Motivation
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5,
Sequences (المتتاليات)
) : ؟ االول المستوى حتى أجدادك عدد كم( الثاني المستوى ؟(حتى( الثالث المستوى ؟(حتى
( اخامس المستوى ؟(حتى
( المستوى ؟(K حتى
Ak = 2K
![Page 6: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/6.jpg)
6,
Train Schedule
??
![Page 7: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/7.jpg)
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In Nature
https://www.youtube.com/watch?v=ahXIMUkSXX0
![Page 8: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/8.jpg)
8,
IQ Tests
1 2 3 4 5
Determine the number of points in the 4th and 5th figure
? ?
![Page 9: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/9.jpg)
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In programing
Any difference between these loops
![Page 10: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/10.jpg)
10,
5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products Part 6: Sequence in Computer Loops and Change of Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 11: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/11.jpg)
11,
Sequences(المتتاليات)
a Sequence is a set of elements written in a row.
Each individual element ak is called a term.
The k in ak is called a subscript or index
am, am+1, am+2, . . . , an
![Page 12: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/12.jpg)
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Finding Terms of Sequences Given by Explicit Formulas
Define sequences a1, a2, a3, . . . and b2, b3, b4, . . . by the following explicit formulas:
Compute the first five terms of both sequences.
الحل ترتيباالفكار وترتيب
< جدا مهماالنماط لمالحظة
![Page 13: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/13.jpg)
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Finding Terms of Sequences Given by Explicit Formulas
Compute the first six terms of the sequence c0 , c1 , c2 , . . .
defined as follows: cj = (−1)j for all integers j ≥
0.
Solution: c0 =(−1)0 =1c1 = (−1)1 = −1 c2 =(−1)2 =1 c3 = (−1)3 = −1 c4 =(−1)4 =1 c5 = (−1)5 = −1
![Page 14: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/14.jpg)
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Finding an Explicit Formula to Fit Given Initial Terms
Both
are
cor
rect
How to prove such formulas of sequences?
Find an explicit formula for a sequence that has the following initial terms:
![Page 15: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/15.jpg)
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5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products Part 6: Sequence in Computer Loops and Change of Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 16: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/16.jpg)
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Summation
Upper limit
Lower limit
index
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Example Let a1 = −2, a2 = −1, a3 = 0, a4 = 1, and a5 = 2. Compute the following:
![Page 18: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/18.jpg)
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When the Terms of a Summation are Given by a Formula
Example
Compute the following summation:
![Page 19: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/19.jpg)
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• Summation to Expanded Form
• Expanded Form to Summation
• Separating Off a Final Term
• Telescoping
Useful Operations
These concepts are very important to understand computer loops
![Page 20: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/20.jpg)
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Summation to Expanded Form
Write the following summation in expanded form:
![Page 21: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/21.jpg)
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Expanded Form to Summation
Express the following using summation notation:
![Page 22: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/22.jpg)
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Separating Off a Final Term and Adding On a Final Term n
Rewrite by separating off the final term.
Write as a single summation.
![Page 23: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/23.jpg)
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Telescoping
Example:
A telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation [wiki].
n
i=1i – (i+1) =(1-2) + (2-3) + . . . + (n –(n+1))
= 1 – (n+1)
=-n
S=0For (i=1;i<=n;i++) S= S+ i-(i+1);
S = -n;
This is very useful in programing:
![Page 24: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/24.jpg)
24,
Telescoping
A telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation [1].
Example: S=0;For (k=1;k<=n;k++) S=S+ 1/k*(k+1);
S = 1- (1/(n+1);
![Page 25: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/25.jpg)
25,
5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products Part 6: Sequence in Computer Loops and Dummy Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 26: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/26.jpg)
26,
Product Notation
![Page 27: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/27.jpg)
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Factorial Notation
0! =12! = 2·1 = 24! = 4·3·2·1 = 246! = 6·5·4·3·2·1 = 720 8! = 8·7·6·5·4·3·2·1 = 40,320
1! =13! =3·2·1=65! = 5·4·3·2·1 = 1207! = 7·6·5·4·3·2·1 = 5,0409! = 9·8·7·6·5·4·3·2·1 = 362,880
![Page 28: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/28.jpg)
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A recursive definition for factorial
Factorial Notation
0! =12! = 2·1 = 24! = 4·3·2·1 = 246! = 6·5·4·3·2·1 = 720 8! = 8·7·6·5·4·3·2·1 = 40,320
1! =13! =3·2·1=65! = 5·4·3·2·1 = 1207! = 7·6·5·4·3·2·1 = 5,0409! = 9·8·7·6·5·4·3·2·1 = 362,880
![Page 29: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/29.jpg)
29,
Computing with Factorials
![Page 30: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/30.jpg)
30,
5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products Part 6: Sequence in Computer Loops and Dummy Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 31: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/31.jpg)
31,
Properties of Summations and Products
Remember to apply these in programing Loops
Which is better in programing?
![Page 32: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/32.jpg)
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Let ak = k + 1 and bk = k - 1 for all integers k. Write each of the following expressions as a single summation or product:
Example
![Page 33: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/33.jpg)
33,
5.1 Sequences
In this lecture:
Part 1: Why we need Sequences (Real-life examples). Part 2: Sequence and Patterns Part 3: Summation: Notation, Expanding & Telescoping Part 4: Product and Factorial Part 5: Properties of Summations and Products
Part 6: Sequence in Computer Loops & Change of Variables
Keywords: Sequences, patterns, Summation, Telescoping, Product, Factorial, Dummy variables,
Mustafa Jarrar: Lecture Notes on Sequences & Mathematical Induction.Birzeit University, Palestine, 2015
Sequences & Mathematical Induction
![Page 34: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/34.jpg)
34,
Change of Variable
Replaced Index by any other symbol (called a dummy variable).
Observe:
Hence:
Also Observe:
![Page 35: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/35.jpg)
35,
Programing Loops
Any difference between these loops
![Page 36: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/36.jpg)
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Change VariablesTransform the following summation by making the specified change of variable.
Change variable j = k+1 For (k=0; k<=6; k++)Sum = Sum + 1/(k+1)
For (k=1; k<=7; k++)Sum = Sum + 1/(k)
![Page 37: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/37.jpg)
37,
Change VariablesTransform the following summation by making the specified change of variable.
For (k=1; k<=n+1; k++)Sum = Sum + k/(n+k)
For (k=0; k<=n; k++)Sum = Sum + (k+1)/(n+k+1)
![Page 38: 1, Mustafa Jarrar Lecture Notes on Sequences & Mathematical Induction. Birzeit University, Palestine, 2015 mjarrar©2015 Sequences & Mathematical Induction.](https://reader036.fdocuments.us/reader036/viewer/2022062315/5697bfa61a28abf838c98579/html5/thumbnails/38.jpg)
38,
Programing Loops
All questions in the exams will be loops
Thus, I suggest:Convert all previous examples into loops and play
with them