The document defines sequences and series. A sequence is an ordered list of elements where order matters. Sequences can be finite or infinite. A series is the sum of the terms of a sequence. Sigma notation is used to represent the sum of terms in a sequence from one index to another. Examples show how to write out the terms of a sequence given a general term formula and how to express a series without sigma notation.
SET
A set is a well defined collection of objects, called the βelementsβ or βmembersβ of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces β{ }β and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as βx ο Aβ
If x is not an element of A, it can be written as βx ο Aβ
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A ο B = {x | x ο A or x ο B}
Let a and b are sets, the intersection of two sets A and B, denoted by A ο B is the set consisting of elements which are in A as well as in B
A ο B = {X | x ο A and x ο B}
If A ο B= ο¦, the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
This powerpoint presentation discusses about the first lesson in Grade 10 Math. It is all about Number Pattern. It also shows the definition, examples and how to find the nth term and general formula.
SET
A set is a well defined collection of objects, called the βelementsβ or βmembersβ of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces β{ }β and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as βx ο Aβ
If x is not an element of A, it can be written as βx ο Aβ
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A ο B = {x | x ο A or x ο B}
Let a and b are sets, the intersection of two sets A and B, denoted by A ο B is the set consisting of elements which are in A as well as in B
A ο B = {X | x ο A and x ο B}
If A ο B= ο¦, the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
This powerpoint presentation discusses about the first lesson in Grade 10 Math. It is all about Number Pattern. It also shows the definition, examples and how to find the nth term and general formula.
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
Part 1 sequence and arithmetic progressionSatish Pandit
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In this presentation, you will learn sequences and arithmetic progression. How to find the terms, common differences, etc. I have given detailed solutions to each problem.
You will learn how to derive patterns in series, also expressing it into summation notation.
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For more information, visit-www.vavaclasses.com
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The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
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Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesarβs dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empireβs birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empireβs society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as βdistorted thinkingβ.
Operation βBlue Starβ is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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2. What is a SEQUENCE?
ο In mathematics, a sequence is an ordered list.
Like a set, it contains members (also
called elements, or terms). The number of
ordered elements (possibly infinite) is called
the length of the sequence. Unlike a set, order
matters, and exactly the same elements can
appear multiple times at different positions in
the sequence. Most precisely, a sequence
can be defined as a function whose domain
is a countable totally ordered set, such as
the natural numbers.
Source:
Wikipedia:http://en.wikipedia.org/wiki/Sequence
3. INFINITE SEQUENCE
ο An infinite sequence is a function with domain the set of
natural numbers π = 1, 2, 3, β¦ β¦ β¦ .
ο For example, consider the function "πβ defined by
π π = π2 π = 1, 2, 3, β¦ β¦
Instead of the usual functional notation π π , for
sequences we usually write
π π = π2
That is, a letter with a subscript, such as π π, is used to
represent numbers in the range of a sequence. For the
sequence defined by π π = π2,
π1 = 12
= 1
π2 = 22
= 4
π3 = 32
= 9
π4 = 42
= 16
4. General or nth Term
ο A sequence is frequently defined by giving its
range. The sequence on the given example
can be written as
1, 4, 9, 16, β¦ β¦ β¦ , π2
, β¦ β¦
Each number in the range of a sequence
is a term of the sequence, with π π the nth term
or general term of the sequence. The formula
for the nth term generates the terms of a
sequence by repeated substitution of counting
numbers for π.
5. FINITE SEQUENCE
ο A finite sequence with π terms is a
function with domain the set of natural
numbers 1, 2,3, β¦ β¦ , π
ο For example, 2, 4, 6, 8, 10 is a finite
sequence with 5 terms where π π = 2π, for
π = 1,2,3,4,5. In contrast, 2,4,6,8,10, β¦ β¦ is an
infinite sequence where π π = 2π, for π =
1,2,3,4,5 β¦ β¦.
6. Sample Problems
1. Write the first five terms of the infinite sequence
with general term π π = 2π β 1.
Answer:
π1 = 2 1 β 1 = 1
π2 = 2 2 β 1 = 3
π3 = 2 3 β 1 = 5
π4 = 2 4 β 1 = 7
π5 = 2 5 β 1 = 9
Thus, the first five terms are 1,3,5,7,9 and the
sequence is
1, 3, 5, 7, 9, β¦ β¦ 2π β 1, β¦ β¦
7. 2. A finite sequence has four terms, and the formula for the
nth term is π₯ π = β1 π 1
2 πβ1. What is the sequence?
Answer:
π₯1 = β1 1 1
21β1 = β1
π₯2 = β1 2 1
22β1 =
1
2
π₯3 = β1 3 1
23β1 = β
1
4
π₯4 = β1 4 1
24β1 =
1
8
Thus the sequence is
β1,
1
2
, β
1
4
,
1
8
8. 3. Find a formula for π π given the first few terms
of the sequence.
a.) 2, 3, 4, 5, β¦
Answer: Each term is one larger than the
corresponding natural number. Thus, we have,
π1 = 2 = 1 + 1
π2 = 3 = 2 + 1
π3 = 4 = 3 + 1
π4 = 5 = 4 + 1
Hence, π π = π + 1
10. SERIES
ο Associated with every sequence, is a
SERIES, the indicated sum of the
sequence.
ο For example, associated with the
sequence 2, 4, 6, 8, 10, is the series
2+4+6+8+10 and associated with the
sequence -1, Β½, -1/4, 1/8, is the series
(-1)+(1/2)+(-1/4)+(1/8).
11. Sigma Summation Notation
ο The Greek letter β (sigma) is often used as a
summation symbol to abbreviate a series.
ο The series 2 + 4 +6 + 8 + 10 which has a general
term π₯ π = 2π, can be written as
π=1
5
π₯ π ππ
π=1
5
2π
and is read as βthe sum of the terms π₯ π or 2π as π
varies from 1 to 5β. The letter π is the index on the
summation while 1 and 5 are the lower and upper
limits of summation, respectively.
12. ο In general, if π₯1, π₯2, π₯3, π₯4 β¦ , π₯ π is a
sequence associated with a series of
π=1
π
π₯ π = π₯1 + π₯2 + π₯3 + π₯4 + β― + π₯ π
15. Break a leg!
1. Find the first four terms and the seventh
term π = 1,2,3, 4 &7 if the general term
of the sequence is π₯ π =
β1 π
π
.
2. Find a formula for π π given the few terms
of the sequence.
1
2
,
1
4
,
1
8
,
1
16
, β¦