1.5 infinite geometric series
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This document discusses infinite geometric series and how to calculate their sums. It defines an infinite geometric series as a series with a first term a and a common ratio r between terms. It presents the formula for calculating the sum of an infinite geometric series as a/(1-r) when the absolute value of r is less than 1. Examples are provided to demonstrate calculating the sums of infinite geometric series using this formula.
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Sue York AAA 2016
The document outlines 7 things that researchers need to know about automation. It discusses how automation has been used for years, can positively impact quality, but also has downsides. The document advocates for more research on how artificial intelligence will be integrated with automation technologies and how automation will impact both quality and quantity of work.
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This document discusses geometric progressions and provides examples of their use. Geometric progressions are sequences where each term is multiplied by a constant ratio. This ratio is called the common ratio. Geometric progressions can model growth patterns like populations over time. Examples shown include smallpox outbreaks, US population growth, and marriage and divorce rates. Geometric progressions are useful for predicting future trends, understanding starting points, and setting goals for companies or other applications involving compound growth or decay over time.
11X1 T10 02 geometric series
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. The document provides examples of finding the common ratio and the general term of geometric series. It also shows how to determine the first term greater than a given value.
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The document provides information about harmonic sequences and the Fibonacci sequence. It discusses how harmonic sequences were first proven in the 14th century but later fell into obscurity until proofs in the 17th century. It also explains that harmonic sequences were used by architects for proportions. An example is given of obtaining harmonic tones on a violin string. It then discusses harmonic means and how to determine the nth term of a harmonic sequence. Finally, it provides background on the Fibonacci sequence, including how it was discovered by Leonardo Fibonacci and that Fibonacci numbers appear in natural patterns.
Arithmetic sequences and arithmetic means
The document defines arithmetic sequences and arithmetic means. An arithmetic sequence is a sequence where each term after the first is equal to the preceding term plus a constant value called the common difference. Formulas are provided for calculating any term in the sequence as well as the sum of the first n terms. Arithmetic means refer to the terms between the first and last term of a sequence. Sample problems demonstrate how to find individual terms, sums of terms, and arithmetic means given the first term and common difference of a sequence.
ARITHMETIC MEAN AND SERIES
Arithmetic Mean and Series
Prepared by:
Darwin Joseph Santos
Mark Joseph Salazar
Cluadine Doma
Maricon Hollon
Randolph Sampang
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The document outlines 7 things that researchers need to know about automation. It discusses how automation has been used for years, can positively impact quality, but also has downsides. The document advocates for more research on how artificial intelligence will be integrated with automation technologies and how automation will impact both quality and quantity of work.
Geometric series
This document discusses geometric progressions and provides examples of their use. Geometric progressions are sequences where each term is multiplied by a constant ratio. This ratio is called the common ratio. Geometric progressions can model growth patterns like populations over time. Examples shown include smallpox outbreaks, US population growth, and marriage and divorce rates. Geometric progressions are useful for predicting future trends, understanding starting points, and setting goals for companies or other applications involving compound growth or decay over time.
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A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. The document provides examples of finding the common ratio and the general term of geometric series. It also shows how to determine the first term greater than a given value.
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Prepared by:
Darwin Joseph Santos
Mark Joseph Salazar
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The document defines a sequence as a set of numbers written in order. It provides the example of a sequence where each term is obtained by adding 1 to the preceding term. This rule is expressed by the equation an = an-1 + 1, where an is the nth term and an-1 is the preceding term. It asks the reader to write the first 5 terms of such a sequence and derive the rule based on the pattern.
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For individual series, the direct method sums all data points and divides by the total number of data points. The shortcut method assumes a mean, calculates the differences from the assumed mean, and finds the mean as the assumed mean plus the sum of the differences divided by the total number of data points.
Sequence powerpoint
The document discusses two arithmetic sequences:
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2) 1,1,2,3,5,8,13,21,... where each term is the sum of the two previous terms.
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This document defines an arithmetic progression as a sequence of numbers where each term is calculated by adding a fixed number (the common difference) to the preceding term. It provides the formulas to calculate the nth term and the sum of the first n terms of an arithmetic progression given the first term, common difference, and n. It also works through examples of finding the 12th term when the 1st term is 2 and common difference is 2, and finding the sum of the first 10 terms when the 1st term is 5 and common difference is 3.
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The geometric mean is a type of average that indicates the central tendency of a set of numbers using their product, as opposed to the arithmetic mean which uses their sum, and it is calculated by taking the nth root of the product of the numbers. The geometric mean is more appropriate than the arithmetic mean for describing proportional growth and ratios, and it has various applications in fields like optics, signal processing, geometry, and finance.
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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.
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3. The document aims to teach students how to identify and order characters, settings, and events in stories.
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The document provides information about arithmetic and geometric sequences and series. It defines sequences, terms, arithmetic sequences, geometric sequences, and gives formulas to find specific terms and sums of sequences. For arithmetic sequences, the formula to find the nth term is an = a1 + d(n-1), where a1 is the first term, d is the common difference, and n is the term number. For geometric sequences, the formula is an = a1rn-1, where a1 is the first term, r is the common ratio, and examples are given to demonstrate using the formulas.
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The document discusses arithmetic and geometric sequences. It provides examples and explanations of key concepts such as common difference, common ratio, formulas for finding specific terms, and calculating series sums. Formulas are derived for finding the nth term in an arithmetic sequence as an = a1 + d(n-1) and the nth term in a geometric sequence as an = a1rn-1. Examples are worked through to demonstrate how to use the formulas and calculate sequence terms and series sums.
Arithmetic Sequence and Series
The document discusses arithmetic sequences and series. Some key points:
1) An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This common difference (d) allows determining the nth term as an = dn + c, where c is the first term minus d.
2) The sum of the first n terms of a finite arithmetic sequence is given by S_n = (n/2)(a_1 + a_n), where a_1 is the first term and a_n is the nth term.
3) The sum of the first n terms of an infinite arithmetic sequence is called the nth partial sum. The partial sums can be represented using summation notation.
Sequences And Series
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
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The document discusses arithmetic progressions (AP), which are lists of numbers where each subsequent term is calculated by adding a fixed number (the common difference) to the previous term. It provides examples of APs with positive, negative, and zero common differences. The general formula for the nth term and sum of the first n terms of an AP are defined. Examples are given to demonstrate calculating specific terms and sums using the formulas.
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This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. It provides the formula for the nth term of an arithmetic sequence as an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. It gives examples of finding specific terms and summarizing sequences. It also discusses the arithmetic mean and arithmetic sum formulas.
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The document contains multiple word problems describing situations that follow arithmetic sequences. Each situation provides the first few terms of the sequence and asks the reader to determine the total or sum based on a given number of terms. The solutions show setting up the arithmetic sequence formula to determine the general nth term and then applying this to calculate the requested total.
Sequence powerpoint 1
This document discusses sequences of events and chronological order. It defines sequence as the order in which events happen, also known as chronological order. It provides examples of signal words that indicate a transition of time such as first, then, and finally. Timelines are presented as a visual way to show sequences and chronological order. The document emphasizes that passages organized in sequence will contain these signal words to guide the order of events.
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This document presents a series of pairs of items and asks "What Comes First?". The pairs include meals (breakfast and lunch), stages of an insect's life cycle (caterpillar and butterfly), tasks (doing the dishes and eating), hygiene routines (getting in the shower and putting clothes on), seasons (fall and winter), numbers (one and two), and life stages (adult and baby). The document concludes by congratulating the reader for completing the assessment.
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The document provides examples and explanations of different types of chronological order and narrative techniques:
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2. It explains "story-within-a-story" as a narrative technique where the story jumps from present events to the past through flashbacks to explain how the characters arrived at their current situation.
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4.6 combinations
This document discusses combinations and how to calculate them. It explains that a combination is an arrangement of objects from a group that considers order irrelevant. The document also provides the fundamental counting principle formula for combinations - nCr = n!/(r!(n-r)!) - and works through several examples of using the formula to calculate the number of possible combinations in different scenarios.
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Sequence and series
The document defines a sequence as a set of numbers written in order. It provides the example of a sequence where each term is obtained by adding 1 to the preceding term. This rule is expressed by the equation an = an-1 + 1, where an is the nth term and an-1 is the preceding term. It asks the reader to write the first 5 terms of such a sequence and derive the rule based on the pattern.
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The document discusses different methods to calculate arithmetic mean from various types of data series.
It explains the direct and shortcut methods to find the arithmetic mean for individual, discrete and continuous data series.
For individual series, the direct method sums all data points and divides by the total number of data points. The shortcut method assumes a mean, calculates the differences from the assumed mean, and finds the mean as the assumed mean plus the sum of the differences divided by the total number of data points.
Sequence powerpoint
The document discusses two arithmetic sequences:
1) 3,6,12,24,48,96,... where each term is double the previous term.
2) 1,1,2,3,5,8,13,21,... where each term is the sum of the two previous terms.
It provides examples of how to determine the pattern/common difference that defines an arithmetic sequence and calculates subsequent terms.
Arithmetic progression
This document defines an arithmetic progression as a sequence of numbers where each term is calculated by adding a fixed number (the common difference) to the preceding term. It provides the formulas to calculate the nth term and the sum of the first n terms of an arithmetic progression given the first term, common difference, and n. It also works through examples of finding the 12th term when the 1st term is 2 and common difference is 2, and finding the sum of the first 10 terms when the 1st term is 5 and common difference is 3.
Geometric Mean
The geometric mean is a type of average that indicates the central tendency of a set of numbers using their product, as opposed to the arithmetic mean which uses their sum, and it is calculated by taking the nth root of the product of the numbers. The geometric mean is more appropriate than the arithmetic mean for describing proportional growth and ratios, and it has various applications in fields like optics, signal processing, geometry, and finance.
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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.
Sequencing events
1. The document describes a sequencing activity for students involving listening to words, phrases, and sentences to identify if they refer to a character, setting, or event.
2. It then provides examples for students to practice sequencing, including sequencing the events in the story of "The Very Hungry Caterpillar".
3. The document aims to teach students how to identify and order characters, settings, and events in stories.
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.
Ppt on sequences and series by mukul sharma
The document provides information about arithmetic and geometric sequences and series. It defines sequences, terms, arithmetic sequences, geometric sequences, and gives formulas to find specific terms and sums of sequences. For arithmetic sequences, the formula to find the nth term is an = a1 + d(n-1), where a1 is the first term, d is the common difference, and n is the term number. For geometric sequences, the formula is an = a1rn-1, where a1 is the first term, r is the common ratio, and examples are given to demonstrate using the formulas.
Maths project work - Arithmetic Sequences
The document discusses arithmetic and geometric sequences. It provides examples and explanations of key concepts such as common difference, common ratio, formulas for finding specific terms, and calculating series sums. Formulas are derived for finding the nth term in an arithmetic sequence as an = a1 + d(n-1) and the nth term in a geometric sequence as an = a1rn-1. Examples are worked through to demonstrate how to use the formulas and calculate sequence terms and series sums.
Arithmetic Sequence and Series
The document discusses arithmetic sequences and series. Some key points:
1) An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This common difference (d) allows determining the nth term as an = dn + c, where c is the first term minus d.
2) The sum of the first n terms of a finite arithmetic sequence is given by S_n = (n/2)(a_1 + a_n), where a_1 is the first term and a_n is the nth term.
3) The sum of the first n terms of an infinite arithmetic sequence is called the nth partial sum. The partial sums can be represented using summation notation.
Sequences And Series
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
Arithmetic, geometric and harmonic progression
This document provides an introduction to arithmetic, geometric, and harmonic progressions. It defines each type of progression and provides examples and formulas for calculating terms, sums, and other properties. Key points covered include the characteristics of an arithmetic progression, formulas for finding sums and identifying terms, geometric progressions and their essential components including starting value, ratio, and number of terms, and the definition and reverse relationship between harmonic and arithmetic progressions.
ARITHMETIC PROGRESSIONS
The document discusses arithmetic progressions (AP), which are lists of numbers where each subsequent term is calculated by adding a fixed number (the common difference) to the previous term. It provides examples of APs with positive, negative, and zero common differences. The general formula for the nth term and sum of the first n terms of an AP are defined. Examples are given to demonstrate calculating specific terms and sums using the formulas.
Arithmetic sequence
This document discusses arithmetic sequences and their properties. It defines an arithmetic sequence as a sequence where the difference between consecutive terms is constant. It provides the formula for the nth term of an arithmetic sequence as an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. It gives examples of finding specific terms and summarizing sequences. It also discusses the arithmetic mean and arithmetic sum formulas.
Arithmetic Sequence Real Life Problems
The document contains multiple word problems describing situations that follow arithmetic sequences. Each situation provides the first few terms of the sequence and asks the reader to determine the total or sum based on a given number of terms. The solutions show setting up the arithmetic sequence formula to determine the general nth term and then applying this to calculate the requested total.
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This document discusses sequences of events and chronological order. It defines sequence as the order in which events happen, also known as chronological order. It provides examples of signal words that indicate a transition of time such as first, then, and finally. Timelines are presented as a visual way to show sequences and chronological order. The document emphasizes that passages organized in sequence will contain these signal words to guide the order of events.
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This document presents a series of pairs of items and asks "What Comes First?". The pairs include meals (breakfast and lunch), stages of an insect's life cycle (caterpillar and butterfly), tasks (doing the dishes and eating), hygiene routines (getting in the shower and putting clothes on), seasons (fall and winter), numbers (one and two), and life stages (adult and baby). The document concludes by congratulating the reader for completing the assessment.
Sequence of events
The document provides examples and explanations of different types of chronological order and narrative techniques:
1. It defines chronological order as events presented in the actual sequence in which they occurred.
2. It explains "story-within-a-story" as a narrative technique where the story jumps from present events to the past through flashbacks to explain how the characters arrived at their current situation.
3. It demonstrates these concepts through examples of ordering events chronologically and using alternating scene formats to signify flashbacks in a story.
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