The document proves that 2n > n^2 for all n > 4 using mathematical induction. It shows:
1) The statement is true for the base case n=5.
2) Assuming the statement is true for some integer k > 4, it proves the statement is also true for k+1.
3) By the principle of mathematical induction, the statement must be true for all integers n > 4.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document discusses absolute value, absolute value equations, and absolute value inequalities. It defines absolute value as the distance from zero on the number line, which is always positive. Absolute value equations account for both positive and negative cases, while absolute value inequalities split into two cases - one for positive values and one for negative values. An example shows how to write the inequalities for both cases of |x| < 4, determine the solution is an intersection of the cases, and represent the solution set as {x | -4 < x < 4}.
Mathematical induction is a method of proof typically used to establish that a property holds for all natural numbers. The proof involves showing that the property holds for the base case, usually n = 1, and then assuming the property holds for some arbitrary natural number k to prove it holds for k + 1. If both steps are true, then the property must hold for all natural numbers by induction.
Mathematical induction has the following steps:
1) Prove that the statement is true for the base case (usually n=1).
2) Assume the statement is true for some integer k.
3) Using the assumption from step 2, prove the statement is true for k+1.
4) By proving the statement true for n=1 and showing that if it is true for k then it is true for k+1, the statement is true for all positive integers n.
The document explains the process of mathematical induction. It begins by proving a statement is true for n=1. It then assumes the statement is true for some value k and proves it is also true for k+1. This establishes the statement is true for all positive integer values of n. Two examples are provided, one proving a statement about n, n+1, n+2 is divisible by 3, and another proving a statement about 33n + 2n+2 is divisible by 5.
The document proves that 2n > n^2 for all n > 4 using mathematical induction. It shows:
1) The statement is true for the base case n=5.
2) Assuming the statement is true for some integer k > 4, it proves the statement is also true for k+1.
3) By the principle of mathematical induction, the statement must be true for all integers n > 4.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document discusses absolute value, absolute value equations, and absolute value inequalities. It defines absolute value as the distance from zero on the number line, which is always positive. Absolute value equations account for both positive and negative cases, while absolute value inequalities split into two cases - one for positive values and one for negative values. An example shows how to write the inequalities for both cases of |x| < 4, determine the solution is an intersection of the cases, and represent the solution set as {x | -4 < x < 4}.
Mathematical induction is a method of proof typically used to establish that a property holds for all natural numbers. The proof involves showing that the property holds for the base case, usually n = 1, and then assuming the property holds for some arbitrary natural number k to prove it holds for k + 1. If both steps are true, then the property must hold for all natural numbers by induction.
Mathematical induction has the following steps:
1) Prove that the statement is true for the base case (usually n=1).
2) Assume the statement is true for some integer k.
3) Using the assumption from step 2, prove the statement is true for k+1.
4) By proving the statement true for n=1 and showing that if it is true for k then it is true for k+1, the statement is true for all positive integers n.
The document explains the process of mathematical induction. It begins by proving a statement is true for n=1. It then assumes the statement is true for some value k and proves it is also true for k+1. This establishes the statement is true for all positive integer values of n. Two examples are provided, one proving a statement about n, n+1, n+2 is divisible by 3, and another proving a statement about 33n + 2n+2 is divisible by 5.
The document discusses the definition and properties of absolute value equations. It defines absolute value as the distance from a number to zero on the real number line. It presents rules for solving absolute value equations, including rewriting an absolute value equation as two separate equations without the absolute value signs, and the property that if the absolute value of an expression is equal to a positive number a, then the expression must equal -a or a. Examples are provided to demonstrate solving absolute value equations using these rules and properties.
- The Earth's crust is made up of tectonic plates that are constantly moving due to convection currents in the mantle.
- There are three types of plate boundaries: divergent where plates separate and new crust is formed, convergent where plates collide and oceanic plates are subducted, and transform where plates slide past each other.
- Evidence for plate tectonics includes magnetic patterns in ocean crust, matching fossils and rock formations on separated continents, and the fit of continental shelves. Plate movements have caused continents like Pangaea to break apart over millions of years.
La planificación tributaria es un proceso que busca invertir de manera eficiente los recursos de un contribuyente para pagar la menor carga impositiva legal posible. Tiene como objetivos definir la mejor alternativa legal para reducir impuestos, estudiar opciones para ahorrar en impuestos, y asegurar el uso efectivo de los recursos de una empresa de acuerdo a su visión. La elusión utiliza estrategias legales como vacíos en la ley para evitar obligaciones tributarias, mientras que la evasión es ilegal y busca ocultar ingres
Changes in education include shifts from traditional single-parent families to various family structures and more children being left alone after school. Socioeconomic status, comprising factors like parents' jobs and education, impacts family background. Technology in classrooms has advanced significantly with computers, internet, tablets, and other devices now commonly used. Schools also face various statistical changes like increased rates of issues around substance abuse, sexuality, bullying, and obesity among students. Educators must address the needs of at-risk student populations including those in poverty, transient students, and those from minority or non-English speaking backgrounds.
Este documento resume un proyecto de evaluación sobre la influencia humana en el relieve y la contaminación de un río. Los estudiantes observan varias influencias humanas en el área como sistemas de riego, cables eléctricos y edificios. Completan un cuestionario sobre residuos, patrimonio cultural y alteración del medio. Concluyen que el río está contaminado y proponen medidas para mejorar la calidad del agua y el hábitat. Enviarán una carta al ayuntamiento con sus hallazgos y recomendaciones.
Este documento presenta 20 reglas de vida propuestas por expertos en ansiedad y estrés para mejorar la calidad de vida. Entre ellas se encuentran hacer pausas durante el trabajo, aprender a decir no, concentrarse en una tarea a la vez, pedir ayuda cuando sea necesario, y entender que uno es responsable de sí mismo y no de los demás. Siguiendo al menos 10 de estas reglas, se puede reducir el estrés y llevar una vida más equilibrada.
En un ejercicio de grupo para una empresa multinacional, se les pidió a tres candidatos que imaginaran lo que les gustaría que dijeran en su velorio. El primero y segundo candidato dieron respuestas convencionales sobre sus logros profesionales y como padre. Sin embargo, el tercer candidato bromeó diciendo que le gustaría que dijeran "¡Mierda, mirá se está moviendo!" para demostrar su optimismo aun después de la muerte. Este último candidato fue contratado por su actitud positiva.
La junta de Castilla y León celebró una sesión de evaluación en la que participaron varios profesores. Durante la sesión, los profesores tomaron algunos acuerdos relacionados con la evaluación y finalizaron la reunión sin otros asuntos pendientes a discutir.
Prueba de la teoría de la deriva continentalEdu 648
El documento describe el mesosaurus, un pequeño reptil que se ha encontrado en sedimentos del Período Pérmico Inferior en el sur de África y América del Sur. En 1911, Alfred Wegener observó que el fósil de este reptil se encontraba en estas dos regiones separadas y utilizó esta evidencia para desarrollar su teoría de la deriva continental, la cual proponía que África y América del Sur habían estado unidas en el pasado como parte de un supercontinente llamado Pangea.
Mrs. Prathima Rudresh taught a class on math skills at PSTTI on December 13th, 2015. The class focused on improving preschool math education. PSTTI works to empower women through education and invited people to join them online or in person to help make a difference in early childhood learning. Contact information was provided for PSTTI, which is located in Bangalore, India.
Este documento presenta una actividad final de semestre para estudiantes de octavo semestre del grupo B titulada "Hagamos lluvia". La actividad busca que los niños de preescolar exploren el proceso de formación de la lluvia mediante una simulación práctica utilizando plantas, un colador, hielos y agua caliente. La actividad se evaluará a través de una escala estimativa y diario del maestro para contrastar las hipótesis iniciales de los niños.
El documento presenta un itinerario de 15 días para un viaje por Cuba que incluye visitas a las ciudades de La Habana, Pinar del Río, Isla de la Juventud, Sancti Spiritus, Camagüey, Holguín y Santiago de Cuba. El itinerario describe las actividades y lugares turísticos a visitar en cada ciudad, así como los hoteles y medios de transporte entre destinos.
Recommendation for successful “the saem” facebook pageFloria Hong
This document provides an analysis and recommendations for improving the Facebook page of The Saem cosmetics brand. It analyzes the Facebook pages of The Saem and competitors like Innisfree and Petitzel. It finds that The Saem's page lacks diverse content related to nature and customer engagement. The document recommends a "Giving Tree Campaign" where liking the page would symbolically plant a tree, with the cover photo updating to show a fuller tree as fans engage. This would promote nature conservation while strengthening the brand relationship through gamified two-way interactions on Facebook.
C# and .NET 4.5 introduced support for asynchronous programming across many common application areas including web access using HttpClient and SyndicationClient, working with files using StorageFile and stream classes, images using MediaCapture and bitmap classes, WCF using synchronous and asynchronous operations, and sockets using the Socket class.
Modulo2.T3.Que necesito para tener un blogProfesorOnline
Para tener un blog, una persona necesita:
1) Alojamiento web gratuito o pago que proporcione espacio para almacenar contenido en internet y un nombre de dominio.
2) Software para crear y administrar el blog, como sistemas de gestión de contenido de código abierto (CMS) como WordPress o plataformas de blogs en línea.
3) Contenido como artículos, fotos y videos para publicar en el blog.
Mohd Mazher Uddin has over 3 years of experience as an Accountant. He has a Post Graduation in MBA and a Bachelor's degree in Commerce. His responsibilities included accounts receivables, accounts payables, cash management, financial reporting, and liaising with banks, customers and suppliers. He is proficient in accounting software like Tally and Microsoft Office applications. He aims to utilize his education and skills in a challenging role in finance and accounts.
LA LUCHA POR LOS DERECHOS.AUTOR: José María Enríquez Sánchez.ISBN: 9788416402861Marcial Pons Argentina
Este documento trata sobre la lucha por los derechos a través de la historia de la inobediencia y sus formas. Se divide en cinco capítulos que exploran formas históricas de inobediencia como la resistencia, la revolución, la revuelta, la contestación y la desobediencia civil. El prólogo introduce el tema examinando diferentes tipos de inobediencia y su posible justificación.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
The document discusses the definition and properties of absolute value equations. It defines absolute value as the distance from a number to zero on the real number line. It presents rules for solving absolute value equations, including rewriting an absolute value equation as two separate equations without the absolute value signs, and the property that if the absolute value of an expression is equal to a positive number a, then the expression must equal -a or a. Examples are provided to demonstrate solving absolute value equations using these rules and properties.
- The Earth's crust is made up of tectonic plates that are constantly moving due to convection currents in the mantle.
- There are three types of plate boundaries: divergent where plates separate and new crust is formed, convergent where plates collide and oceanic plates are subducted, and transform where plates slide past each other.
- Evidence for plate tectonics includes magnetic patterns in ocean crust, matching fossils and rock formations on separated continents, and the fit of continental shelves. Plate movements have caused continents like Pangaea to break apart over millions of years.
La planificación tributaria es un proceso que busca invertir de manera eficiente los recursos de un contribuyente para pagar la menor carga impositiva legal posible. Tiene como objetivos definir la mejor alternativa legal para reducir impuestos, estudiar opciones para ahorrar en impuestos, y asegurar el uso efectivo de los recursos de una empresa de acuerdo a su visión. La elusión utiliza estrategias legales como vacíos en la ley para evitar obligaciones tributarias, mientras que la evasión es ilegal y busca ocultar ingres
Changes in education include shifts from traditional single-parent families to various family structures and more children being left alone after school. Socioeconomic status, comprising factors like parents' jobs and education, impacts family background. Technology in classrooms has advanced significantly with computers, internet, tablets, and other devices now commonly used. Schools also face various statistical changes like increased rates of issues around substance abuse, sexuality, bullying, and obesity among students. Educators must address the needs of at-risk student populations including those in poverty, transient students, and those from minority or non-English speaking backgrounds.
Este documento resume un proyecto de evaluación sobre la influencia humana en el relieve y la contaminación de un río. Los estudiantes observan varias influencias humanas en el área como sistemas de riego, cables eléctricos y edificios. Completan un cuestionario sobre residuos, patrimonio cultural y alteración del medio. Concluyen que el río está contaminado y proponen medidas para mejorar la calidad del agua y el hábitat. Enviarán una carta al ayuntamiento con sus hallazgos y recomendaciones.
Este documento presenta 20 reglas de vida propuestas por expertos en ansiedad y estrés para mejorar la calidad de vida. Entre ellas se encuentran hacer pausas durante el trabajo, aprender a decir no, concentrarse en una tarea a la vez, pedir ayuda cuando sea necesario, y entender que uno es responsable de sí mismo y no de los demás. Siguiendo al menos 10 de estas reglas, se puede reducir el estrés y llevar una vida más equilibrada.
En un ejercicio de grupo para una empresa multinacional, se les pidió a tres candidatos que imaginaran lo que les gustaría que dijeran en su velorio. El primero y segundo candidato dieron respuestas convencionales sobre sus logros profesionales y como padre. Sin embargo, el tercer candidato bromeó diciendo que le gustaría que dijeran "¡Mierda, mirá se está moviendo!" para demostrar su optimismo aun después de la muerte. Este último candidato fue contratado por su actitud positiva.
La junta de Castilla y León celebró una sesión de evaluación en la que participaron varios profesores. Durante la sesión, los profesores tomaron algunos acuerdos relacionados con la evaluación y finalizaron la reunión sin otros asuntos pendientes a discutir.
Prueba de la teoría de la deriva continentalEdu 648
El documento describe el mesosaurus, un pequeño reptil que se ha encontrado en sedimentos del Período Pérmico Inferior en el sur de África y América del Sur. En 1911, Alfred Wegener observó que el fósil de este reptil se encontraba en estas dos regiones separadas y utilizó esta evidencia para desarrollar su teoría de la deriva continental, la cual proponía que África y América del Sur habían estado unidas en el pasado como parte de un supercontinente llamado Pangea.
Mrs. Prathima Rudresh taught a class on math skills at PSTTI on December 13th, 2015. The class focused on improving preschool math education. PSTTI works to empower women through education and invited people to join them online or in person to help make a difference in early childhood learning. Contact information was provided for PSTTI, which is located in Bangalore, India.
Este documento presenta una actividad final de semestre para estudiantes de octavo semestre del grupo B titulada "Hagamos lluvia". La actividad busca que los niños de preescolar exploren el proceso de formación de la lluvia mediante una simulación práctica utilizando plantas, un colador, hielos y agua caliente. La actividad se evaluará a través de una escala estimativa y diario del maestro para contrastar las hipótesis iniciales de los niños.
El documento presenta un itinerario de 15 días para un viaje por Cuba que incluye visitas a las ciudades de La Habana, Pinar del Río, Isla de la Juventud, Sancti Spiritus, Camagüey, Holguín y Santiago de Cuba. El itinerario describe las actividades y lugares turísticos a visitar en cada ciudad, así como los hoteles y medios de transporte entre destinos.
Recommendation for successful “the saem” facebook pageFloria Hong
This document provides an analysis and recommendations for improving the Facebook page of The Saem cosmetics brand. It analyzes the Facebook pages of The Saem and competitors like Innisfree and Petitzel. It finds that The Saem's page lacks diverse content related to nature and customer engagement. The document recommends a "Giving Tree Campaign" where liking the page would symbolically plant a tree, with the cover photo updating to show a fuller tree as fans engage. This would promote nature conservation while strengthening the brand relationship through gamified two-way interactions on Facebook.
C# and .NET 4.5 introduced support for asynchronous programming across many common application areas including web access using HttpClient and SyndicationClient, working with files using StorageFile and stream classes, images using MediaCapture and bitmap classes, WCF using synchronous and asynchronous operations, and sockets using the Socket class.
Modulo2.T3.Que necesito para tener un blogProfesorOnline
Para tener un blog, una persona necesita:
1) Alojamiento web gratuito o pago que proporcione espacio para almacenar contenido en internet y un nombre de dominio.
2) Software para crear y administrar el blog, como sistemas de gestión de contenido de código abierto (CMS) como WordPress o plataformas de blogs en línea.
3) Contenido como artículos, fotos y videos para publicar en el blog.
Mohd Mazher Uddin has over 3 years of experience as an Accountant. He has a Post Graduation in MBA and a Bachelor's degree in Commerce. His responsibilities included accounts receivables, accounts payables, cash management, financial reporting, and liaising with banks, customers and suppliers. He is proficient in accounting software like Tally and Microsoft Office applications. He aims to utilize his education and skills in a challenging role in finance and accounts.
LA LUCHA POR LOS DERECHOS.AUTOR: José María Enríquez Sánchez.ISBN: 9788416402861Marcial Pons Argentina
Este documento trata sobre la lucha por los derechos a través de la historia de la inobediencia y sus formas. Se divide en cinco capítulos que exploran formas históricas de inobediencia como la resistencia, la revolución, la revuelta, la contestación y la desobediencia civil. El prólogo introduce el tema examinando diferentes tipos de inobediencia y su posible justificación.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
The document outlines the steps of mathematical induction to prove that 2n is greater than n^2 for all integers n greater than 4. It shows:
1) The basis step that proves this is true for n=5
2) The induction assumption that the statement holds true for some integer k greater than 4
3) The induction step that proves the statement holds true for k+1 if it is true for k
4) By the principle of mathematical induction, the statement must be true for all integers greater than 4.
The document discusses mathematical induction. It proves that for all integers n greater than or equal to 1, 1 + 2 + 3 + ... + n is less than or equal to n^2. It does this by showing the statement holds for n=1, and assuming it is true for some integer k implies it is also true for k+1.
The document discusses mathematical induction and provides examples of using it to prove statements. It introduces the concept of mathematical induction, which involves three steps: 1) proving the statement is true for n=1, 2) assuming it is true for an integer k, and 3) proving it is true for k+1. The document then works through two examples - proving that n(n+1)(n+2) is divisible by 3, and proving that 3^n + 2^(n+2) is divisible by 5.
The document describes the steps of mathematical induction. It includes:
Step 1: Prove the result is true for the first term, usually n = 1.
Step 2: Assume the result is true for an integer k.
Step 3: Prove the result is true for k + 1, using the assumption from Step 2.
Step 4: Conclude that since the result is true for n = 1 by Step 1, and true for n = k + 1 by Step 3, it is true for all positive integers n by induction.
The example provided works through an induction proof for the sum of odd integers from 1 to 2n - 1.
This document summarizes several theorems related to mathematical induction and principles of mathematical induction (PMI). It begins by defining the standard PMI, which states that if a set S satisfies the properties that 1 is in S and if n is in S then n+1 is in S, then S is equal to the set of natural numbers. It then provides proofs of several theorems using mathematical induction. Finally, it provides examples of using variations of PMI and strong induction to prove additional theorems and properties.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1(n+2) is divisible by 3 for all positive integers n. It shows the basis step for n=1 and inductive step, assuming true for n=k and proving for n+1. The second example proves 33n+2n+2 is divisible by 5 for all n, following the same process of basis and inductive steps.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1nn+2 is divisible by 3 for all positive integers n. It shows the basis step for n=1 and inductive step, assuming true for n=k and proving for n+1. Secondly, it proves 33n+2n+2 is divisible by 5 for all n, again using a basis step and inductive step. The document demonstrates the key steps of mathematical induction.
The document discusses mathematical induction and provides examples of using it to prove statements. Specifically, it shows how to prove that expressions like nn+1nn+2 and 33n+2n+2 are divisible by 3 and 5 for all positive integer values of n. The proof involves showing the base case is true, assuming the statement holds for an integer k, and then showing it also holds for k+1 based on the assumption for k.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1nn+2 is divisible by 3 for all positive integers n. It shows the basis step for n=1 and inductive step, assuming true for n=k and proving for n+1. Secondly, it proves 33n+2n+2 is divisible by 5 for all n, again using a basis step and inductive step. The document demonstrates the key steps of mathematical induction.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1nn+2 is divisible by 3 for all positive integers n. It shows the basis step for n=1 is true, assumes the statement is true for n=k, and proves it is true for n=k+1. The second example proves that the expression 33n+2n+2 is divisible by 5 for all n, following the same induction structure.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
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The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
12. Proof:
1 1 1 1 1 1 1
1 2 1 2 2 2
22 3 k 12 2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
13. Proof:
1 1 1 1 1 1 1
1 2 1 2 2 2
22 3 k 12 2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
14. Proof:
1 1 1 1 1 1 1
1 2 1 2 2 2
22 3 k 12 2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
15. Proof:
1 1 1 1 1 1 1
1 2 1 2 2 2
22 3 k 12 2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
1
2
k 1
16. Proof:
1 1 1 1 1 1 1
1 2 1 2 2 2
22 3 k 12 2 3 k k 12
1 1
2
k k 12
k 1 k
2
2
k k 1
2
k 2 k 1
2
k k 1
2
k2 k 1
2
k k 1 k k 1
2 2
k k 1
2
k k 1
2
1
2
k 1
1 1 1 1
1 2 2 2
2 3 k 12
k 1
17. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
18. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1
19. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
20. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
21. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
22. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
Proof:
23. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
24. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
25. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
4
2
26. (ii) A sequence is defined by;
a1 2 an1 2 an for n 1
Show that an 2 for n 1
Test: n = 1 a1 2 2
A n k a k 2
P n k 1 ak 1 2
Proof:
ak 1 2 ak
22
4
2
ak 1 2
27. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
28. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1
29. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
30. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
31. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
32. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
33. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1 x y
2 k k
34. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1 x y
2 k k
xk y k
10
35. iii The sequences xn and yn are defined by;
xn y n 2 xn y n
x1 5, y1 2 xn1 , yn1
2 xn y n
Prove xn yn 10 for n 1
Test: n = 1 x1 y1 52
10
A n k xk yk 10
P n k 1 xk 1 yk 1 10
Proof:
xk yk 2 xk yk
xk 1 yk 1 x y
2 k k
xk y k
10
xk 1 yk 1 10
36. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
37. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2
38. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2
L.H .S a1
1
39. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
40. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S
41. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S
L.H .S a2
1
42. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
43. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
44. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
k 1 k
1 5 1 5
A n k 1 & n k ak 1 & ak
2 2
45. (iv) The Fibonacci sequence is defined by;
a1 a2 1 an1 an an1 for n 1
n
1 5
Prove that an for n 1
2
Test: n = 1 and n =2 1
1 5
L.H .S a1 R.H .S
2
1
1.62
L.H .S R.H .S 2
1 5
L.H .S a2 R.H .S
2
1
2.62
L.H .S R.H .S
k 1 k
1 5 1 5
A n k 1 & n k ak 1 & ak
2 2
k 1
1 5
P n k 1 ak 1
2