Pre-Cal 40S Slides January 18, 2008
•
0 likes•428 views
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.
Report
Share
Report
Share
Download to read offline
Recommended
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.
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.
Pre-Cal 40S Slides May 30, 2007
The document provides examples of geometric sequences and series, including finding terms in a sequence given the first term and common ratio, as well as calculating the sum of terms in a geometric series using sigma notation. It also defines arithmetic series and shows how to calculate the sum of terms in an arithmetic sequence using the formula for the sum to the nth term of an arithmetic series. Examples are given of both geometric and arithmetic sequences and series.
Applied Math 40S Slides May 30, 2007
Introduction to sequences, arithmetic, geometric and others. Recursively and implicitly definitions. Using the graphing calculator find the value of any term in a sequence.
Per5 sequences
The document discusses sequences and summations. It defines a sequence as a function from a subset of natural numbers to a set S. A sequence is denoted {an} where an is the image of the integer n. Formulas can be used to describe sequences. Finite sequences are also called strings. Summations represent the sum of terms in a sequence from an index m to n. The variable under the summation sign is called the index of summation. Friedrich Gauss developed a formula to simplify calculating summations. There can also be double or triple summations corresponding to nested loops.
Sequence formulas direct and recursive
The document discusses sequence formulas, including direct and recursive formulas. It provides examples of sequences defined by direct formulas like the sequence where the nth term is defined as An = 9n + 3. It then explains that a recursive formula defines each term of a sequence using preceding terms, and must state the initial term(s). It gives examples of calculating sequence terms using general formulas and recursion equations. Finally, it briefly mentions the famous Fibonacci sequence and its applications.
Sequences
A sequence is a set of terms that follow a pattern or rule. If the sequence continues indefinitely, it is called an infinite sequence, otherwise it is finite. The rule allows you to write an expression for the nth term (un). Common examples include arithmetic sequences where the difference is constant, and geometric sequences where each term is obtained by multiplying the previous term by a fixed value. A recurrence relation defines each term in terms of the preceding ones. To solve problems involving sequences, you determine the expression for successive terms and find sums by adding terms up using the Greek letter sigma. Exam questions may involve finding individual terms, sums, or showing sums are divisible.
A19-5 solving quads by formula
This document provides instructions for solving quadratic equations using the quadratic formula. It begins by asking what the quadratic formula is, then lists the steps to use it: write the equation in standard form with terms of highest degree to lowest, identify the a, b, and c coefficients, and plug these values into the quadratic formula to solve for x.
Recommended
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.
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.
Pre-Cal 40S Slides May 30, 2007
The document provides examples of geometric sequences and series, including finding terms in a sequence given the first term and common ratio, as well as calculating the sum of terms in a geometric series using sigma notation. It also defines arithmetic series and shows how to calculate the sum of terms in an arithmetic sequence using the formula for the sum to the nth term of an arithmetic series. Examples are given of both geometric and arithmetic sequences and series.
Applied Math 40S Slides May 30, 2007
Introduction to sequences, arithmetic, geometric and others. Recursively and implicitly definitions. Using the graphing calculator find the value of any term in a sequence.
Per5 sequences
The document discusses sequences and summations. It defines a sequence as a function from a subset of natural numbers to a set S. A sequence is denoted {an} where an is the image of the integer n. Formulas can be used to describe sequences. Finite sequences are also called strings. Summations represent the sum of terms in a sequence from an index m to n. The variable under the summation sign is called the index of summation. Friedrich Gauss developed a formula to simplify calculating summations. There can also be double or triple summations corresponding to nested loops.
Sequence formulas direct and recursive
The document discusses sequence formulas, including direct and recursive formulas. It provides examples of sequences defined by direct formulas like the sequence where the nth term is defined as An = 9n + 3. It then explains that a recursive formula defines each term of a sequence using preceding terms, and must state the initial term(s). It gives examples of calculating sequence terms using general formulas and recursion equations. Finally, it briefly mentions the famous Fibonacci sequence and its applications.
Sequences
A sequence is a set of terms that follow a pattern or rule. If the sequence continues indefinitely, it is called an infinite sequence, otherwise it is finite. The rule allows you to write an expression for the nth term (un). Common examples include arithmetic sequences where the difference is constant, and geometric sequences where each term is obtained by multiplying the previous term by a fixed value. A recurrence relation defines each term in terms of the preceding ones. To solve problems involving sequences, you determine the expression for successive terms and find sums by adding terms up using the Greek letter sigma. Exam questions may involve finding individual terms, sums, or showing sums are divisible.
A19-5 solving quads by formula
This document provides instructions for solving quadratic equations using the quadratic formula. It begins by asking what the quadratic formula is, then lists the steps to use it: write the equation in standard form with terms of highest degree to lowest, identify the a, b, and c coefficients, and plug these values into the quadratic formula to solve for x.
Algebra 2 unit 12
This document introduces summation notation and how to evaluate series expressed in sigma notation. It explains that a series represents the sum of terms in a sequence and defines partial sums. Summation notation uses the capital sigma (Σ) symbol to represent the sum of terms defined by a rule. The document provides examples of writing series in summation notation and evaluating them. It also derives formulas for finding the sums of constant, linear, quadratic, and other common series without having to expand them out.
Alg2 lesson 2.4
This document discusses finding the slope and y-intercept of a line given its equation in slope-intercept form. It provides examples of writing equations of lines given their slopes and points, or two points the line passes through. It also discusses writing the equation of a line perpendicular to another line given a point on the perpendicular line.
Théory of 4 twins, distribution of prime numbers
This document discusses prime numbers of the form 6n + -1, where n is a natural number. It presents a list of prime numbers and non-prime numbers less than 100 of this form, showing that the non-prime numbers are products of two prime factors that are both of the form 6n + -1. It also categorizes pairs of these numbers as different types of "twins" based on whether they are both prime or if one is a multiple of the other. Finally, it presents a method for determining all numbers of the form 6n + -1 using a prime number P and the expression P + (Px4) + (Px2).
Alg2 lesson 2 4
This document discusses finding the equation of a line given different information. It covers:
1) Finding the slope (m=4/3) and y-intercept (b=-3) of the line 6x - 3y = 7 using the slope-intercept form of a line (y=mx+b).
2) Writing the equation of a line containing the point (1,2) with slope -1/2 using the point-slope form (y-y1=m(x-x1)).
3) Writing the equation of a line containing the points (-2,2) and (3,7) by finding the slope between the points and using the point-slope form
Math aruthmetic geometric series
The document discusses summation formulas for arithmetic and geometric series. It provides the formulas to calculate the sum of a finite arithmetic series as the mean of the first and last term multiplied by the number of terms. For a geometric series with a common ratio r not equal to 1 and divisor not equal to 0, the formula is the first term divided by 1 minus the common ratio. An example calculates the 30th term of an arithmetic series with initial term of 4 and difference of 3 between terms. It also finds a rule for the nth term of a geometric sequence given the first two terms.
11.1 Sequences and Series
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
Algebra 2 unit 12.1
This document provides an overview of mathematical patterns and sequences. It defines key terms like sequence, term, recursive formula, and explicit formula. Examples demonstrate how to find sequence terms using recursive and explicit rules, write rules for linear and quadratic patterns, and calculate terms of fractal sequences generated by repeated iterations. The document also illustrates sequence patterns with examples like the Fibonacci sequence, flushing water in a toilet, and the Cantor set fractal.
Unit 6.1
The document provides examples for writing linear equations in slope-intercept form (y=mx+b) given different representations of a line:
1) Given the slope (m) and y-intercept (b), write the equation.
2) Given a graph of the line, calculate the slope and y-intercept to write the equation.
3) Given the coordinates of two points on the line, calculate the slope and use it to write the equation.
4) Given an equation and a point that the line passes through, solve for the missing slope (m) or y-intercept (b) value.
3rd Test - Writing The Equation Of A Line
This document discusses finding the equation of a line from different information provided. It addresses finding the equation from a graph by using two points to determine the slope and y-intercept. It also discusses finding the equation of a parallel line given the equation of another line and a point, as well as finding the equation of a perpendicular line given the same information. The document suggests writing out the process for each method to find the line equation and reflecting on how these methods could have helped with previous assignments.
Sequences and series
The document provides information about different types of infinite sequences, including explicitly defined sequences, recursively defined sequences, arithmetic sequences, harmonic sequences, and geometric sequences. It gives examples and formulas for finding terms, sums, and analyzing properties of each type of sequence. Key points covered include the definitions of explicit and recursive sequences, the formulas for the nth term and partial sum of arithmetic and geometric sequences, and examples of finding terms and analyzing sequences.
130543763770937500
This document defines arithmetic progression and arithmetic series. It explains that arithmetic progression is a sequence with a constant difference between terms, and can be represented by an explicit formula relating the first term, common difference, and term number. Arithmetic series is a series with a constant difference between terms, and its sum can be found using a formula multiplying the number of terms by the average of the first and last terms. It provides examples of applying these concepts and formulas.
Aug 28th
This document outlines an algebra lesson where students will use deductive reasoning to prove algebraic properties. The lesson includes simplifying expressions as a warmup, then working on problems from the textbook involving finding patterns in tables to relate terms involving n to Sn and Sn+1-Sn.
maths ppt on exponents, playing with no. & graphs
The document contains multiple presentations on math topics including:
1) Exponents and laws of exponents with examples of multiplying and dividing terms with the same base.
2) Cartesian coordinate system explained with an example point P(4,2) defining the x and y coordinates.
3) Properties of two and three digit numbers in terms of place value and examples of number patterns.
4) Tests for divisibility of numbers by 2, 3, 5, 9, and 10.
Pre-Cal 40S Slides January 16, 2008
The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
5.5 parallel and perpendicular lines (equations) day 1
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.
Finding slope
This is a ppt for learning how to calculate the slope of a line within a line graph. This ppt works for about a 8th t 9th grade level.
Arithmetic Sequence
The document defines sequences and series, and discusses arithmetic sequences in particular. An arithmetic sequence is a sequence where each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. The document provides examples of finding the common difference using the formulas d=an+1-an and d=an-a1. It also gives examples of finding specific terms of an arithmetic sequence given information like the first term, common difference, and nth term.
Identifying slope and y intercept
The document discusses the equation of a line, y=mx+b, and its key components of slope and y-intercept. It defines slope as the rate of rise or fall of a line and provides the formula to calculate it between two points. It also defines the y-intercept as the point where the line crosses the y-axis. The document provides examples of lines with their slopes and y-intercepts calculated from points and used to write the equation of the line.
Writing and Graphing slope intercept form
The document discusses graphing and writing equations in slope-intercept form (y=mx+b). It explains that the slope (m) is the coefficient and the y-intercept (b) is the constant. It provides examples of graphing lines by plotting the y-intercept and counting up/right by the slope. It also demonstrates writing equations given the slope and/or a point, including finding missing values like the y-intercept.
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.
Pre-Cal 40S June 5, 2009
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.
More Related Content
What's hot
Algebra 2 unit 12
This document introduces summation notation and how to evaluate series expressed in sigma notation. It explains that a series represents the sum of terms in a sequence and defines partial sums. Summation notation uses the capital sigma (Σ) symbol to represent the sum of terms defined by a rule. The document provides examples of writing series in summation notation and evaluating them. It also derives formulas for finding the sums of constant, linear, quadratic, and other common series without having to expand them out.
Alg2 lesson 2.4
This document discusses finding the slope and y-intercept of a line given its equation in slope-intercept form. It provides examples of writing equations of lines given their slopes and points, or two points the line passes through. It also discusses writing the equation of a line perpendicular to another line given a point on the perpendicular line.
Théory of 4 twins, distribution of prime numbers
This document discusses prime numbers of the form 6n + -1, where n is a natural number. It presents a list of prime numbers and non-prime numbers less than 100 of this form, showing that the non-prime numbers are products of two prime factors that are both of the form 6n + -1. It also categorizes pairs of these numbers as different types of "twins" based on whether they are both prime or if one is a multiple of the other. Finally, it presents a method for determining all numbers of the form 6n + -1 using a prime number P and the expression P + (Px4) + (Px2).
Alg2 lesson 2 4
This document discusses finding the equation of a line given different information. It covers:
1) Finding the slope (m=4/3) and y-intercept (b=-3) of the line 6x - 3y = 7 using the slope-intercept form of a line (y=mx+b).
2) Writing the equation of a line containing the point (1,2) with slope -1/2 using the point-slope form (y-y1=m(x-x1)).
3) Writing the equation of a line containing the points (-2,2) and (3,7) by finding the slope between the points and using the point-slope form
Math aruthmetic geometric series
The document discusses summation formulas for arithmetic and geometric series. It provides the formulas to calculate the sum of a finite arithmetic series as the mean of the first and last term multiplied by the number of terms. For a geometric series with a common ratio r not equal to 1 and divisor not equal to 0, the formula is the first term divided by 1 minus the common ratio. An example calculates the 30th term of an arithmetic series with initial term of 4 and difference of 3 between terms. It also finds a rule for the nth term of a geometric sequence given the first two terms.
11.1 Sequences and Series
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
Algebra 2 unit 12.1
This document provides an overview of mathematical patterns and sequences. It defines key terms like sequence, term, recursive formula, and explicit formula. Examples demonstrate how to find sequence terms using recursive and explicit rules, write rules for linear and quadratic patterns, and calculate terms of fractal sequences generated by repeated iterations. The document also illustrates sequence patterns with examples like the Fibonacci sequence, flushing water in a toilet, and the Cantor set fractal.
Unit 6.1
The document provides examples for writing linear equations in slope-intercept form (y=mx+b) given different representations of a line:
1) Given the slope (m) and y-intercept (b), write the equation.
2) Given a graph of the line, calculate the slope and y-intercept to write the equation.
3) Given the coordinates of two points on the line, calculate the slope and use it to write the equation.
4) Given an equation and a point that the line passes through, solve for the missing slope (m) or y-intercept (b) value.
3rd Test - Writing The Equation Of A Line
This document discusses finding the equation of a line from different information provided. It addresses finding the equation from a graph by using two points to determine the slope and y-intercept. It also discusses finding the equation of a parallel line given the equation of another line and a point, as well as finding the equation of a perpendicular line given the same information. The document suggests writing out the process for each method to find the line equation and reflecting on how these methods could have helped with previous assignments.
Sequences and series
The document provides information about different types of infinite sequences, including explicitly defined sequences, recursively defined sequences, arithmetic sequences, harmonic sequences, and geometric sequences. It gives examples and formulas for finding terms, sums, and analyzing properties of each type of sequence. Key points covered include the definitions of explicit and recursive sequences, the formulas for the nth term and partial sum of arithmetic and geometric sequences, and examples of finding terms and analyzing sequences.
130543763770937500
This document defines arithmetic progression and arithmetic series. It explains that arithmetic progression is a sequence with a constant difference between terms, and can be represented by an explicit formula relating the first term, common difference, and term number. Arithmetic series is a series with a constant difference between terms, and its sum can be found using a formula multiplying the number of terms by the average of the first and last terms. It provides examples of applying these concepts and formulas.
Aug 28th
This document outlines an algebra lesson where students will use deductive reasoning to prove algebraic properties. The lesson includes simplifying expressions as a warmup, then working on problems from the textbook involving finding patterns in tables to relate terms involving n to Sn and Sn+1-Sn.
maths ppt on exponents, playing with no. & graphs
The document contains multiple presentations on math topics including:
1) Exponents and laws of exponents with examples of multiplying and dividing terms with the same base.
2) Cartesian coordinate system explained with an example point P(4,2) defining the x and y coordinates.
3) Properties of two and three digit numbers in terms of place value and examples of number patterns.
4) Tests for divisibility of numbers by 2, 3, 5, 9, and 10.
Pre-Cal 40S Slides January 16, 2008
The document discusses sequences and provides examples of arithmetic and geometric sequences. It defines key terms like sequence, arithmetic sequence, common difference, geometric sequence, and common ratio. Formulas are given for finding the nth term in an arithmetic sequence and a geometric sequence. Examples are shown for calculating the common difference and finding a specific term in an arithmetic sequence.
5.5 parallel and perpendicular lines (equations) day 1
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.
Finding slope
This is a ppt for learning how to calculate the slope of a line within a line graph. This ppt works for about a 8th t 9th grade level.
Arithmetic Sequence
The document defines sequences and series, and discusses arithmetic sequences in particular. An arithmetic sequence is a sequence where each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. The document provides examples of finding the common difference using the formulas d=an+1-an and d=an-a1. It also gives examples of finding specific terms of an arithmetic sequence given information like the first term, common difference, and nth term.
Identifying slope and y intercept
The document discusses the equation of a line, y=mx+b, and its key components of slope and y-intercept. It defines slope as the rate of rise or fall of a line and provides the formula to calculate it between two points. It also defines the y-intercept as the point where the line crosses the y-axis. The document provides examples of lines with their slopes and y-intercepts calculated from points and used to write the equation of the line.
Writing and Graphing slope intercept form
The document discusses graphing and writing equations in slope-intercept form (y=mx+b). It explains that the slope (m) is the coefficient and the y-intercept (b) is the constant. It provides examples of graphing lines by plotting the y-intercept and counting up/right by the slope. It also demonstrates writing equations given the slope and/or a point, including finding missing values like the y-intercept.
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.
What's hot (20)
5.5 parallel and perpendicular lines (equations) day 1
5.5 parallel and perpendicular lines (equations) day 1
Similar to Pre-Cal 40S Slides January 18, 2008
Pre-Cal 40S June 5, 2009
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.
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.
sequence and series.docx
The document defines and describes different types of sequences, including arithmetic, harmonic, and geometric sequences. It also discusses the convergence properties of sequences, defining convergent, divergent, and oscillating sequences. Some techniques for evaluating limits of convergent sequences are presented, including using continuous function representations and properties of polynomials.
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.
MATHS PPT 2 (2).pptx
A sequence is an ordered list of numbers where each subsequent term is obtained by applying a specific rule to the previous term(s). A sequence can be finite, with a limited number of terms, or infinite and continue indefinitely. A series is the sum of the terms in a sequence, denoted by Σan where an is the nth term. There are different types of series such as arithmetic where a constant is added to each term, geometric where a constant is multiplied to each term, and harmonic where the reciprocal of each term is added. An arithmetic progression is a specific type of sequence where each term is obtained by adding a fixed constant, called the common difference, to the preceding term.
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.
20 sequences x
The document defines sequences and provides examples of convergent and non-convergent sequences. It states that a sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Examples are provided of sequences with patterns like odd numbers and square numbers, and sequences without obvious patterns. It further explains that a sequence converges if the limit of its terms is finite, and provides examples of sequences that converge, like terms approaching 0, -1, and 1. Criteria for determining if a sequence converges is given in terms of bounding all terms within any interval of the limit.
20 sequences x
The document defines sequences and provides examples of convergent and non-convergent sequences. It states that a sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Examples are provided of sequences with patterns like odd numbers and square numbers, and sequences without obvious patterns. It further explains that a sequence converges if the limit of its terms is finite, and provides examples of sequences that converge, like terms approaching 0, -1, and 1. Criteria for determining if a sequence converges is given in terms of bounding all terms within any interval of the limit.
Pre-Cal 20S January 21, 2009
The document discusses arithmetic sequences and provides examples of finding terms in arithmetic sequences. It defines an arithmetic sequence as a list of numbers generated by adding a common difference to successive terms. It provides the recursive and implicit definitions of an arithmetic sequence and explains how to find the common difference and the nth term of a sequence given its first term and common difference.
Sequences and series
The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.
math.pptx
This document provides information about sequences and series in mathematics. It defines a sequence as an ordered set of real numbers denoted by a1, a2, a3, etc. where an represents the general term. Sequences can be finite or infinite. A sequence is said to converge to a limit l if the terms get arbitrarily close to l as n increases. Bounded sequences are those where there exists a number k such that an is always less than k. Monotonic sequences either always increase or decrease. A series is an infinite sum of terms of a sequence. The convergence, divergence or oscillation of a series depends on whether the partial sums sn tend to a finite limit, infinity, or fail to have a unique limit as
Sequences and series
The document defines sequences and series in precalculus. A sequence is a function with positive integers as its domain. A series represents the sum of the terms of a sequence. The document provides examples of arithmetic and geometric sequences, and defines their associated series. It also discusses infinite geometric series and harmonic sequences. Examples are given to identify sequences and series, determine sequence terms, and identify types of sequences.
Arithmetic sequences and series
This document introduces arithmetic sequences and series. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. The nth term of an arithmetic sequence can be written as a formula of a, the first term, and d, the common difference. An arithmetic series is the sum of terms in an arithmetic sequence, which can be calculated using the sigma notation and the formula for the sum of an arithmetic sequence. Examples are provided to illustrate finding terms and sums of arithmetic sequences and series.
Arithmetic progression
The document discusses arithmetic progression (A.P.), which is a series where the difference between consecutive terms is constant. It provides examples like the summation of natural numbers from 1 to n. The key aspects are:
1) An A.P. has a first term, a common difference between terms, and a formula to calculate any term as the first term plus the product of the common difference and the term number minus 1.
2) The formula for summing the first n natural numbers is n(n+1)/2.
3) By analogy, the formula for summing an A.P. from its first term a to its nth term is n(2a+(n-1)d)/
Arithmetic sequences and series[1]
The document discusses arithmetic and geometric sequences. It defines a sequence as a set of numbers in order. Sequences can be finite or infinite. Arithmetic sequences have a common difference between terms, while geometric sequences multiply each term by a constant ratio. Formulas are provided to calculate specific terms and the sum of terms in arithmetic and geometric sequences.
(678215997) neethutext
This document provides an introduction to sequences and series. It begins with definitions of sequences, finite and infinite sequences, and series. It then covers topics like arithmetic progressions, geometric progressions, and harmonic progressions. It provides formulas for the nth term and sum of terms for arithmetic and geometric progressions. It also defines arithmetic mean and geometric mean between terms in progressions. The document aims to help secondary students understand key concepts related to sequences and series.
Sequences
This document discusses different types of sequences:
1. Arithmetic sequences are sequences where the difference between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common difference.
2. Geometric sequences are sequences where the ratio between consecutive terms is constant. The nth term can be calculated using a formula involving the first term and the common ratio.
3. Examples are provided to demonstrate how to calculate the nth term and sum of terms for both arithmetic and geometric sequences. Formulas are given for finding individual terms and summing finite and infinite sequences.
3.-SEQUENCES-AND-SERIES-THEORY.hhsssspdf
Jshshsusjwusbzhzhssjzisjsjshsbzjjznsnsjzhzjswnsnjszjzhsnjskzzjbzjsskzjhsnsnsnsjsnsnsjsnnzzjjzjzzjzjzjjzjzjzjjzjzjznsbsbszbzbzjzjzjzjjssjjzjzsjzjzjzjzjsjsjjzjzzjjzzjjzzjjzzjzj
Similar to Pre-Cal 40S Slides January 18, 2008 (20)
More from Darren Kuropatwa
Behind Their Eyes v1
Behind Their Eyes - making thinking visible is not enough
Walk into any classroom and watch the breakneck pace at which teachers are working hard to help students learn. Mind you, if we don’t uncover what students are thinking while learning, they may be running down the wrong path. OK, so we need ways to make student thinking visible. Seeing their thinking is important, but we also need to create the time and space for teachers to absorb, reflect, and act on what their students thinking reveals. This workshop shares strategies both for making student thinking visible and for creating time and space for teachers to meaningfully act on what they learn about what’s going on behind their eyes.
Leading Change v1
“If you really want to understand something, try changing it.” - Kurt Lewin
As the Director of Learning for a school division made up of 18 schools, my job is to help lead the largest change initiative ever undertaken in our school community. One of the most important, difficult, messy things any school leader does is lead change. While we can learn from the change leadership of others, copying their work most often leads to failure. Success is more likely to come from adapting others work to our own context. In this workshop I share the journey we’ve undertaken collectively in our schools; how we developed a shared vision, cultivated collaborative cultures, maintained a focus on deep learning, and wrestle with the nuances of accountability. Informed by the latest research on change management in education, we also model strategies for fostering deep learning conversations in your schools. We’ll engage in some deeper learning conversations together and take back a wealth of ideas you can adapt to your own context. Developing collaborative cultures is careful and precise work that has profound impact when carried out well. So how do you do that? Come, let’s learn together. Good people are important, but good cultures are moreso.
Providing Permission To Wonder v2.1
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore inquiry as a pedagogical stance and the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants will leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
Things That Suck About Digital Citizenship v1
The document is a presentation about digital citizenship given by Darren Kuropatwa at the Building Learning Communities Conference in Boston, MA in July 2017. It discusses the importance of digital citizenship and responding to adversity with persistent kindness. It provides examples of digital citizenship issues and scenarios for discussion. It encourages participants to think about their own digital footprint and how to be good digital citizens.
Digital Storytelling for Deeper Learning v1
Presented at the Riding the Wave Conference in Gimli, Manitoba. May 2017.
In two words, you remember the whole story: glass slipper, sour grapes, cold porridge. You remember more than facts, you recall relationships & deeper connections between characters. Some of the powerful ways we leverage digital for deeper learning includes challenging sources of information (fake news), exploring bias (developing empathy through multiple perspectives), and creating powerful feedback loops that foster deeper learning.
Powerful narratives, in a word or two, bring to mind a wealth of ideas & relationships; more than just facts. How can we find stories that make our teaching sticky and help kids find, and more importantly tell, stories that make learning stick? This workshop will equip teachers with the skills & knowledge to foster deeper learning across the curriculum by intentionally leveraging digital tools to foster deeper learning.
Tales of Learning and the Gifts of Footprints v4.2
This document appears to be a presentation about digital learning and storytelling. It discusses shifting from compliance to care, private to public learning, and consumer to participatory models. It addresses what digital storytellers look like and principles of learning including starting where students are, learning being done by and for students, students talking about learning, having learning targets, and feedback. It encourages generosity, sharing tales of learning, and giving the gifts of footprints.
The Fourth Screen v4.2
Presented at the Richmond District Conference, Feb 2017.
A series of stories woven together to start a conversation with middle and high school students, teachers, and parents about living our lives on and offline (on The Fourth Screen) more thoughtfully.
This talk focuses primarily on the ideas of Empathy, Empowerment & Persistent Kindness and shares resources teachers can use to lead these sorts of conversations with their own students.
Making Student Thinking Visible v4
Slides to support a master class on making student thinking visible through practical hands-on activities and structured around Dylan Wiliam's work on formative assessment and active learning. Held at the BYTE Conference 2017 in Portage la Prairie, Manitoba.
Learning is at BYTE 2017
A group of educators from the BYTE Conference 2017 (Build Your Teaching Experience) share their ideas about learning as a series of visual metaphors they found on their phones.
Leveraging Digital for Deeper Learning
Slides to support a Keynote presentation to Early Years and Middle Years teachers in Hanover School Division on 3 February 2017.
We Learn Through Stories at PRIZMAH17
The document discusses storytelling and how it can be used as a tool for learning. It suggests that storytelling allows students to think in metaphors and learn through stories. It provides examples of how digital tools like QR codes and apps can be used to incorporate storytelling into the classroom. It also outlines some rules of thumb for using storytelling, such as personalizing tasks to students' experiences, collaborating on group projects, and getting feedback from both inside and outside the classroom.
Providing Permission to Wonder v3
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
The Fourth Screen v4.1
This document contains multiple sections on topics related to technology and its impact on society, including how the internet allows information to be easily shared but also persist indefinitely, issues around online privacy and bullying, and ways for parents to support their children's safe and responsible internet use. The document advocates for empowering youth and promoting kindness both online and off.
Making Student Thinking Visible v3.7
Slides to support a master class on making student thinking visible through practical hands-on activities and structured around Dylan Wiliam's work on formative assessment and active learning.
Learning is at AUHSD
A group of educators from the Anderson Union High School & Redding School Districts and share their ideas about learning as a series of visual metaphors.
Providing Permission to Wonder v2
In a world where knowledge is more a verb than a noun how do we foster deep learning in our students? Good questions cause thinking. Unfortunately, many of the questions regularly asked in classrooms focus on knowledge as a noun. This presentation will explore the effective use of thinking and learning tools in the classroom. We will work together to model teaching practices that lead to students co-constructing a networked (real world) rather than hierarchical (artificial) understanding of their world regardless of grade level or discipline.
Participants leave this workshop with a toolkit of research based questioning and thinking strategies they can begin using with their students tomorrow.
The Fourth Screen v4
This document contains a collection of images, quotes, and short passages on topics related to online communities, sharing, and empowerment through technology. The snippets discuss how the internet allows information to be easily shared, encourages learning, and can help empower victims of bullying. The overarching theme is about the positive impact community and connection through online platforms can provide.
Learning is at BLC16
A group of educators at the Building Learning Communities Conference share their ideas about learning as a series of visual metaphors.
We Learn Through Stories v4 (master class)
Slides to support a master class at the Building Learning Communities Conference in Boston, MA. 18 July 2016.
How can we make learning sticky using powerful storytelling frameworks that tap into peoples' emotions? How do we involve all students in creating digital content that doesn't also create hours of content for teachers to assess? This interactive session will showcase Digital Storytelling activities teachers can use in class tomorrow! Document student learning & foster reflective ways for students to share their learning. 1st: we play! Then we'll discuss how to practically adapt these ideas, make them your own, and figure out what sort of infrastructure needs to be in place to support these kinds of powerful learning experiences. We’ll learn how to exercise your students' & your own creativity muscles and share simple strategies for collecting & publishing student work.
Deep Learning Design: the middle ring
Slides in support of a professional learning day for administrators in Hanover School Division focused on developing a common language & understanding of Deep Learning Design.
More from Darren Kuropatwa (20)
Tales of Learning and the Gifts of Footprints v4.2
Tales of Learning and the Gifts of Footprints v4.2
Recently uploaded
Educational Technology in the Health Sciences
Plenary presentation at the NTTC Inter-university Workshop, 18 June 2024, Manila Prince Hotel.
Wound healing PPT
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Temple of Asclepius in Thrace. Excavation results
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
How to Download & Install Module From the Odoo App Store in Odoo 17
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
How to Manage Reception Report in Odoo 17
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
مصحف أحرف الخلاف للقراء العشرةأعد أحرف الخلاف بالتلوين وصلا سمير بسيوني غفر الله له
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰
How to Setup Default Value for a Field in Odoo 17
In Odoo, we can set a default value for a field during the creation of a record for a model. We have many methods in odoo for setting a default value to the field.
HYPERTENSION - SLIDE SHARE PRESENTATION.
IT WILL BE HELPFULL FOR THE NUSING STUDENTS
IT FOCUSED ON MEDICAL MANAGEMENT AND NURSING MANAGEMENT.
HIGHLIGHTS ON HEALTH EDUCATION.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Recently uploaded (20)
SWOT analysis in the project Keeping the Memory @live.pptx
SWOT analysis in the project Keeping the Memory @live.pptx
Juneteenth Freedom Day 2024 David Douglas School District
Juneteenth Freedom Day 2024 David Douglas School District
How to Download & Install Module From the Odoo App Store in Odoo 17
How to Download & Install Module From the Odoo App Store in Odoo 17
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
مصحف القراءات العشر أعد أحرف الخلاف سمير بسيوني.pdf
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
A Free 200-Page eBook ~ Brain and Mind Exercise.pptx
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
Pre-Cal 40S Slides January 18, 2008
- 1. Listen to the music ... the geometric sequences music .... Intermezzo
- 2. Evan Ma - Sequences and Series http://youtube.com/watch?v=M1WzkWy3n74
- 3. The Story of Young Gauss ... Photo Source: Karl Gauss (1777–1855) to be continued ...
- 5. SEQUENCE SERIES
- 6. 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
- 7. (a) What is the sum of the integers from 1 to 5000?
- 8. (b) What is the sum of all multiples of 7 between 1 & 5000? (c) What is the sum of all integers from 1 to 5000 inclusive that are not multiples of 7?
- 13. Sigma Notation: A shorthand way to write a series. Example: means (2(1) -3) + (2(2) -3) + (2(3) -3) + (2(4) -3) = -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
- 14. Find the value of:
- 16. Given a geometric sequence in which and , what is the value of ?