Pre-Cal 40S June 5, 2009
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The document discusses geometric series and their formulas for calculating the sum of terms. It defines a geometric series as having a first term (a) and a common ratio (r) between terms. The formula to calculate the sum of the first n terms is provided as (a(1-r^n))/(1-r). An example of finding the term with a value of 27 in a geometric series is given. The conditions for a geometric series to converge or diverge are outlined based on the absolute value of the common ratio r.
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Pre-Cal 40S June 4, 2009
The document discusses arithmetic and geometric sequences and series. It provides the formulas to find the nth term of an arithmetic sequence and geometric sequence. Examples are given to find specific terms. It also discusses how to calculate the sum of terms in an arithmetic series and geometric series using the summation notation. Formulas are given for the sum to the nth term of an arithmetic series and geometric series.
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This document provides instruction on writing equations of lines using different forms: slope-intercept, point-slope, two-point, and intercept forms. Examples are given for writing equations of lines when given characteristics like slope, points, or intercepts. The last section presents an application example of using line equations to determine if two sets of bones found in an excavation site are parallel.
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Supervised by Prof. Dr.Tazibur Rahman
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Construction of C3V character table
GROUP THEORY
CONSTRUCTING CHARACTER TABLE IS FOLLOWED BY 4 STEPS through orthogonality rule
STEP 1 : FIND THE NUMBER OF IRRs
Number of IRs = Number of classes.- In C3v
there is 3 classes so Г1,Г2 Г3
STEP 2: FIND OUT THE DIMENSIONS
Sum of the squares of the dimensions of IRRs = Order of the Group
We have to identify a set of 3 positive integers (I1 I2 I3 dimensions of IRRs) which satisfy this condition
The only value of I which satisfy this condition are 1,1,2 so that I12 = I22
SO 3 IRRs of C3v ,two are 1-D and one is 2-D
STEP 3 : FIND character of two 1-D IRRs
In every point group is 1-D IRR who characters are equal to 1 .this IRRs is called totally symmetric IRR
Thus we have
Which satisfy the rule sum of the square of the characters of all operations in any IRR is equal to the order of the group
FIND characters of another 1-D IRRs Conditions
All the characters of this IRRs equal to +1 or -1
Also IRR must be Orthogonal to Г1
Г1 has six +1 as characters of the sym operations 1 for E ; 2 (1) for C3 ; 3 (1) for σv
The characters of Г2 is Orthogonal to Г1 so it has three +1 and three -1
For E in 1-D is +1 ; for 2 C3 in 1-D is +1 ; FOR 3 σV is -1
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The document provides information about solving various types of inequalities, including:
- Linear inequalities can be expressed and solved using inequality notation, set notation, interval notation, and graphically. When multiplying or dividing by a negative number, the inequality sign must be reversed.
- Non-linear inequalities like quadratics, polynomials, and rationals can be solved by identifying intervals where the expression is positive or negative, based on its zeros. Rational inequalities also require excluding values that make the denominator equal to zero.
- The solution set of any inequality can be written as an interval using correct notation.
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The document discusses the gamma function for negative numbers. It begins by providing background on the gamma function and its typical definition for positive integers and real numbers. The behavior for negative numbers is more complex, as the typical integral definition does not converge. The paper aims to develop a formula for the gamma function for different negative numbers using mathematical inference and Stirling's asymptotic formula. It presents formulas for some known negative numbers like -0.5 and -1.5, and then develops a general formula to calculate the gamma function for any negative number. It applies this formula to examples like -0.2, -0.6, -1.1 and presents the results in a table.
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A geometric series is the sum of the terms of a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio. The document provides an example of a geometric series where the first term is 1 and the ratio between successive terms is 2, giving the series 1, 2, 4, 8, 16, etc. whose sum is 31. The document then states that it will derive the general formula for the sum of a geometric series by using a technique similar to that used for arithmetic series.
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1. The document instructs readers to fold a square piece of paper into quarters and record the resulting areas in a table after each fold.
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This document provides instruction on writing equations of lines using different forms: slope-intercept, point-slope, two-point, and intercept forms. Examples are given for writing equations of lines when given characteristics like slope, points, or intercepts. The last section presents an application example of using line equations to determine if two sets of bones found in an excavation site are parallel.
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1) A geometric series is the expression of the sum of terms in a geometric sequence. The sum of a finite geometric series can be calculated using the formula Sn = a(1 - r^n)/(1 - r) where a is the first term, r is the common ratio, and n is the number of terms.
2) Geometric series converge when the absolute value of the common ratio r is less than 1, and diverge when the absolute value of r is greater than 1.
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The document provides a study guide for a Power Point test covering topics from Thursday's test as well as formulas. It includes warm-up problems on ratios, unit conversions, and simplifying expressions. It reviews finding slopes and determining if lines are parallel. It discusses writing equations of lines given a point and the equation of a parallel or perpendicular line. Sample problems are worked through demonstrating how to find the equation of a line passing through a point that is parallel or perpendicular to another given line.
A Short Study of Galois Field
Create by Hazrat Ali Naim(session 2010-2011 )
Supervised by Prof. Dr.Tazibur Rahman
Dhaka University
Dhaka-1000
Construction of C3V character table
GROUP THEORY
CONSTRUCTING CHARACTER TABLE IS FOLLOWED BY 4 STEPS through orthogonality rule
STEP 1 : FIND THE NUMBER OF IRRs
Number of IRs = Number of classes.- In C3v
there is 3 classes so Г1,Г2 Г3
STEP 2: FIND OUT THE DIMENSIONS
Sum of the squares of the dimensions of IRRs = Order of the Group
We have to identify a set of 3 positive integers (I1 I2 I3 dimensions of IRRs) which satisfy this condition
The only value of I which satisfy this condition are 1,1,2 so that I12 = I22
SO 3 IRRs of C3v ,two are 1-D and one is 2-D
STEP 3 : FIND character of two 1-D IRRs
In every point group is 1-D IRR who characters are equal to 1 .this IRRs is called totally symmetric IRR
Thus we have
Which satisfy the rule sum of the square of the characters of all operations in any IRR is equal to the order of the group
FIND characters of another 1-D IRRs Conditions
All the characters of this IRRs equal to +1 or -1
Also IRR must be Orthogonal to Г1
Г1 has six +1 as characters of the sym operations 1 for E ; 2 (1) for C3 ; 3 (1) for σv
The characters of Г2 is Orthogonal to Г1 so it has three +1 and three -1
For E in 1-D is +1 ; for 2 C3 in 1-D is +1 ; FOR 3 σV is -1
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Based on Maths chapter 1 of class 8 it consists of every topic and a good explanation. Please read the full ppt. It will also teach you how to design a ppt also. so reading these is a good way of gaining knowledge. It consists of every topic in the book and can be used a a teaching purpose also.
Inequalities
The document provides information about solving various types of inequalities, including:
- Linear inequalities can be expressed and solved using inequality notation, set notation, interval notation, and graphically. When multiplying or dividing by a negative number, the inequality sign must be reversed.
- Non-linear inequalities like quadratics, polynomials, and rationals can be solved by identifying intervals where the expression is positive or negative, based on its zeros. Rational inequalities also require excluding values that make the denominator equal to zero.
- The solution set of any inequality can be written as an interval using correct notation.
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The document discusses writing equations of parallel and perpendicular lines, including identifying whether lines are parallel or perpendicular based on their slopes. It provides examples of writing equations of parallel and perpendicular lines given a point and the slope or line of another line. Guidance problems are also included to practice these skills.
doc
This document presents an algorithm for finding a mod 4-flow in graphs. It begins with definitions of concepts like sparks (graphs with no 3-edge coloring), mod k-flows, and reducible configurations. It proves that the 2-sum of two sparks is also a spark. The algorithm orients the complement of a feasible set of weight 2 edges to find a mod 4-flow. It concludes that a graph has a mod 4-flow if every component of the complement has an even number of vertices with a certain labeling, and that sparks cannot satisfy this.
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This document discusses writing equations of lines. It explains that a linear equation can be written in the form of y=mx+b, where m is the slope and b is the y-intercept. It provides examples of writing the equation of a line when given a slope and point, and when given two points, by finding the slope and using one of the points to solve for b and write the full equation in y=mx+b form.
Gamma function for different negative numbers and its applications
The document discusses the gamma function for negative numbers. It begins by providing background on the gamma function and its typical definition for positive integers and real numbers. The behavior for negative numbers is more complex, as the typical integral definition does not converge. The paper aims to develop a formula for the gamma function for different negative numbers using mathematical inference and Stirling's asymptotic formula. It presents formulas for some known negative numbers like -0.5 and -1.5, and then develops a general formula to calculate the gamma function for any negative number. It applies this formula to examples like -0.2, -0.6, -1.1 and presents the results in a table.
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This document outlines learning objectives for integration and antiderivatives. The objectives are:
1) Find antiderivatives of functions
2) Represent antiderivatives
3) Use basic antidifferentiation techniques and integration rules
The document then provides examples of finding antiderivatives using basic integration rules like recognizing patterns when integrating polynomial functions. It also covers the substitution method for integration, where a function is substituted with another function to simplify the integral.
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The document discusses sequences and series in mathematics. It defines arithmetic and geometric sequences, and introduces the concepts of series notation using sigma notation to succinctly write out the terms of a sequence. It provides examples of finding sums of arithmetic sequences and series, as well as an example problem solving for a term in a geometric sequence given initial terms.
Applied 40S May 28, 2009
The document discusses sequences and their properties. It defines arithmetic and geometric sequences, and explains how to find the common difference or ratio, as well as the nth term of a sequence. It provides examples of solving for these values. The document also discusses using a calculator to work with sequences and lists some practice problems.
Applied Math 40S June 2 AM, 2008
The document discusses arithmetic sequences, including defining them recursively or through an implicit linear equation, finding the common difference and nth term, and examples of determining terms and sums of arithmetic sequences given initial terms or the values of a and d. It also provides homework problems involving identifying arithmetic sequences, writing terms, and calculating sums of sequences.
Pre-Cal 40S Slides June 3, 2008
The document discusses Karl Gauss, a mathematician from the 18th century. It provides some mathematical examples involving geometric sequences and series. It also defines key concepts like arithmetic series and sigma notation for writing series in shorthand. The document references additional online resources about Gauss and sequences and series.
Gg
The document defines key concepts related to sequences and series:
- An infinite sequence is a function with positive integers as its domain. Geometric sequences have a common ratio between consecutive terms.
- The nth term of a geometric sequence is given by an = a1rn-1, where a1 is the first term and r is the common ratio.
- A geometric series is the sum of terms of an infinite geometric sequence. If the common ratio r satisfies |r|<1, the sum of the infinite series can be calculated as a1/(1-r).
- Examples demonstrate calculating individual terms, sums of finite sequences, and sums of infinite series using the given formulas.
Class_Powerpoint_Sequences_Arithmetic_and_Geometric_with_Series_Finite_and_In...
1. The document discusses arithmetic and geometric sequences.
2. Arithmetic sequences are defined by adding a common difference to get the next term, while geometric sequences multiply by a common ratio.
3. Examples are provided of finding terms in arithmetic and geometric sequences using formulas.
Pre-Cal 40S Slides June 2, 2008
This document defines sequences and provides information about arithmetic and geometric sequences. It discusses the recursive and implicit definitions of sequences. For arithmetic sequences, it explains how to find the common difference and the nth term using the common difference and the first term. For geometric sequences, it similarly explains how to find the common ratio and the nth term using the common ratio and first term.
Arithmetic and geometric mean
This document introduces sequences and series in mathematics. It defines a sequence as a set of numbers written in a particular order, with the n-th term written as un. A series is the sum of terms in a sequence. An arithmetic progression has terms where each new term is obtained by adding a constant difference to the preceding term. The n-th term of an arithmetic progression is a + (n - 1)d, where a is the first term and d is the common difference. A geometric progression multiplies each new term by a constant ratio r to obtain the next term, with the n-th term written as arn-1. Formulas are provided for finding the n-th term, sum of terms,
11.1 Sequences and Series
This document defines sequences and series and provides examples of how to write terms of sequences and evaluate partial sums of series. It discusses writing sequences as functions with the natural numbers as the domain and the term values as the range. Examples are provided of finding the next term in a sequence and using Desmos to list terms. The document also defines convergent and divergent sequences, introduces summation notation for writing series, and provides properties and rules for manipulating summations including evaluating finite series.
Arithmetic And Geometric Progressions
This document introduces sequences and series, focusing on arithmetic and geometric progressions. It provides:
1) Definitions of sequences, series, and the differences between them. Arithmetic progressions have a common difference between terms, while geometric progressions multiply the previous term by a common ratio.
2) Formulas for calculating terms and sums of arithmetic and geometric progressions. The sum of an arithmetic progression can be expressed in terms of the first term, last term, number of terms, and common difference.
3) Examples of using the formulas to find specific terms and sums of progressions. Practice with these types of problems is important to master the techniques.
Ap gp
This document introduces arithmetic and geometric progressions. It defines a sequence as a set of numbers written in a particular order. A series is the sum of the terms in a sequence. An arithmetic progression is a sequence where each new term is obtained by adding a constant difference to the preceding term. The sum of an arithmetic progression can be found using the formula: the sum of the first n terms is equal to one-half n times the quantity of two times the first term plus (n - 1) times the common difference.
9.3 Geometric Sequences
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
5_Week-5_ PRE CALCULUS SIGMA NOTATION Series.pdf
The document defines and provides examples of sigma notation, which is used to represent the sum of terms in a sequence. Sigma notation indicates the sum of terms from an index running from a lower limit to an upper limit. Examples show how to write out series using sigma notation and evaluate the sums. Properties of sigma notation allow evaluating sums using algebraic rules.
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- 1. Infinite Geometric Series iphone to infinity for righty's by flickr user KIT
- 2. Given a geometric sequence in which and , what is the value of ?
- 3. Series: The sum of numbers in a sequence to a particular term in a sequence. Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms. Artithmetic Series: The sum of numbers in an arithmetic sequence given by is the sum to the nth term n is the quot;rankquot; of the nth term a is the first term in the sequence d is the common difference
- 4. Sigma Notation: A shorthand way to write a series. Example: 4 ∑(2n - 3) means (2(1) -3) + (2(2) -3) + (2(3) -3) + (2(4) -3) n=1 = -1 + 1 + 3 + 5 =8 Σ is capital sigma (from the greek alphabet); means sum subscript n = 1 means quot;start with n = 1 and evaluate (2n - 3)quot; superscript 4 means keep evaluating (2n - 3) for successive integral values of n; stop when n = 4; then add all the terms (2n - 3) is the implicit definition of the sequence
- 5. Series: The sum of numbers in a sequence to a particular term in a sequence. Example: denotes the sum of the first 5 terms. denotes the sum of the first n terms. Geometric Series: The sum of numbers in an geometric sequence given by or is the sum to the nth term n is the quot;rankquot; of the nth term a is the first term in the sequence d is the common difference
- 6. or
- 8. Given the geometric sequence in which and r = , which term has a value of 27? Find the sum of the first 5 terms.
- 9. Given the geometric sequence in which and r = , which term has a value of 27? Find the sum of the first 5 terms.
- 11. Infinite Geometric Series iphone to infinity for righty's by flickr user KIT
- 13. Infinite Geometric Series Why is that the formula?
- 14. CONVERGENT SERIES 0 < |r| <1 DIVERGENT SERIES |r| > 1
- 15. Find the infinite sum for a geometric series given: a = 12 r= 2 3