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1) The document uses mathematical induction to prove several formulas. 2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1). 3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.

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5.4 mathematical induction

The document discusses the method of mathematical induction. It is used to verify infinitely many related statements without checking each one individually. As an example, it examines the statement that the sum of the first n odd numbers equals n^2 for all natural numbers n. It shows the base case of this statement is true, and if the statement is true for an arbitrary n, it must also be true for n+1. Therefore, by the principle of mathematical induction, the statement is true for all natural numbers n.

Principle of mathematical induction

The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.

Mathematical induction

Mathematical induction is a method of proof that can be used to prove that a statement is true for all positive integers. It involves two steps: 1) proving the statement is true for the base case, usually n = 1, and 2) assuming the statement is true for an integer k and using this to prove the statement is true for k + 1. Examples are provided to demonstrate how to use mathematical induction to prove statements such as the sum of the first n positive integers equalling n(n+1)/2 and that 7n - 1 is divisible by 6 for all positive integers n.

Mathematical induction

1) Mathematical induction is a method of proof that can be used to prove statements for all positive integers. It involves showing that a statement is true for n=1, and assuming it is true for an integer k to prove it is true for k+1.
2) The document provides an example using mathematical induction to prove the formula Sn = n(n+1) for the sum of the first n even integers.
3) Finite differences are used to determine if a sequence has a quadratic model by seeing if the second differences are constant. The example finds the quadratic model n^2 for the sequence 1, 4, 9, 16, 25, 36.

Mathematical induction by Animesh Sarkar

The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.

mathematical induction

This document contains lecture notes on the principle of mathematical induction. It includes examples, exercises, and solutions working through proofs using induction for equations numbered 23-2 through 23-25. Each section is labeled with the relevant equation and includes steps to prove or solve problems related to that equation inductively.

INTEGRATION BY PARTS PPT

The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.

5.1 anti derivatives

The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.

5.4 mathematical induction

The document discusses the method of mathematical induction. It is used to verify infinitely many related statements without checking each one individually. As an example, it examines the statement that the sum of the first n odd numbers equals n^2 for all natural numbers n. It shows the base case of this statement is true, and if the statement is true for an arbitrary n, it must also be true for n+1. Therefore, by the principle of mathematical induction, the statement is true for all natural numbers n.

Principle of mathematical induction

The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.

Mathematical induction

Mathematical induction is a method of proof that can be used to prove that a statement is true for all positive integers. It involves two steps: 1) proving the statement is true for the base case, usually n = 1, and 2) assuming the statement is true for an integer k and using this to prove the statement is true for k + 1. Examples are provided to demonstrate how to use mathematical induction to prove statements such as the sum of the first n positive integers equalling n(n+1)/2 and that 7n - 1 is divisible by 6 for all positive integers n.

Mathematical induction

1) Mathematical induction is a method of proof that can be used to prove statements for all positive integers. It involves showing that a statement is true for n=1, and assuming it is true for an integer k to prove it is true for k+1.
2) The document provides an example using mathematical induction to prove the formula Sn = n(n+1) for the sum of the first n even integers.
3) Finite differences are used to determine if a sequence has a quadratic model by seeing if the second differences are constant. The example finds the quadratic model n^2 for the sequence 1, 4, 9, 16, 25, 36.

Mathematical induction by Animesh Sarkar

The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.

mathematical induction

This document contains lecture notes on the principle of mathematical induction. It includes examples, exercises, and solutions working through proofs using induction for equations numbered 23-2 through 23-25. Each section is labeled with the relevant equation and includes steps to prove or solve problems related to that equation inductively.

INTEGRATION BY PARTS PPT

The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.

5.1 anti derivatives

The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.

CMSC 56 | Lecture 11: Mathematical Induction

Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Metric space

This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.

Recursion DM

The document discusses recursion and recursively defined sequences, functions, and algorithms. It provides examples of recursively defined sequences like powers of 2 and the Fibonacci sequence. Recursively defined functions are given, such as factorials and Fibonacci numbers. Recursive algorithms solve problems by reducing them to smaller instances of the same problem, like the Euclidean algorithm to find the greatest common divisor or a recursive Fibonacci algorithm. While recursive solutions are often shorter and more elegant, iterative algorithms using loops are usually more efficient.

Binomial Theorem

The binomial theorem provides a formula for expanding binomial expressions of the form (a + b)^n. It states that the terms of the expansion are determined by binomial coefficients. Pascal's triangle is a mathematical arrangement that shows the binomial coefficients and can be used to determine the coefficients in a binomial expansion. The proof of the binomial theorem uses mathematical induction to show that the formula holds true for any positive integer value of n.

Integral calculus

1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a

Number Theory - Lesson 1 - Introduction to Number Theory

This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes

Limit of functions

1) One-sided limits describe the value a function approaches as the input gets closer to a number from the left or right.
2) The limit of a function exists if and only if the one-sided limits are equal as the input approaches the number.
3) Limits at infinity describe the value a function approaches as the input increases without bound toward positive or negative infinity.

21 monotone sequences x

The document defines and discusses monotone sequences. A sequence {an} is defined as increasing if an < an+1, non-decreasing if an ≤ an+1, decreasing if an > an+1, and non-increasing if an ≥ an+1. Methods for determining if a sequence is monotone include the difference method, ratio method, and derivative method. Bounded and eventually monotone sequences are shown to converge according to the monotone sequence convergence theorem.

Number theory

This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.

Radical expressions

The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.

Relations in Discrete Math

A relation is a set of ordered pairs that shows a relationship between elements of two sets. An ordered pair connects an element from one set to an element of another set. The domain of a relation is the set of first elements of each ordered pair, while the range is the set of second elements. Relations can be represented visually using arrow diagrams or directed graphs to show the connections between elements of different sets defined by the relation.

number theory.ppt

This document discusses key concepts in number theory including divisibility, greatest common divisors, least common multiples, prime and composite numbers, relative primality, modular arithmetic, factorials, and applications. It defines these terms and provides examples. Greatest common divisors are the largest integers that divide two numbers. Least common multiples are the smallest integers divisible by two numbers. Prime numbers have only two factors and composite numbers are multiples of primes. Relative primality means two numbers have no common prime factors. Modular arithmetic uses the remainder of a division. Factorials are the product of integers up to a given number. Applications include cryptography.

Recurrence relations

This document discusses recurrence relations and their use in defining sequences. It introduces key concepts like recurrence relations, initial conditions, explicit formulas, and solving recurrence relations using techniques like backtracking or finding the characteristic equation. As examples, it examines the Fibonacci sequence and linear homogeneous recurrence relations of varying degrees.

Permutation and combination

1) The document discusses permutations and combinations, which are ways of arranging or selecting items from a group.
2) A permutation is an arrangement of items that considers order, while a combination disregards order.
3) Formulas are provided to calculate the number of permutations and combinations for a given number of items selected from a larger set.

Integration by partial fraction

this presentation covers the topic of integration by partial fraction...let me know if u finfd this helpful

Math induction principle (slides)

The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.

10-Sequences and summation.pptx

The document discusses sequences and summations. It provides examples and definitions of different types of sequences such as arithmetic and geometric sequences. It also discusses recurrence relations, which express a term in a sequence based on prior terms. Examples are provided to demonstrate finding terms of sequences given a recurrence relation. The document is a lecture on discrete mathematics concepts related to sequences.

Complex numbers And Quadratic Equations

This ppt is on the above mentioned topic with clear and interactive explainations and some examples too!

Remainder theorem

The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.

Limits

The document discusses the concept of limits. It explains that as the number of sides of a polygon increases, the area of the polygon approximates the area of the circle it is inscribed in, and the limit of the polygon's area is equal to the area of the circle. It also examines the limit of a function as x approaches 2 from both sides, and defines some fundamental rules of limits, such as the constant rule, sum rule, and multiplication rule. Finally, it outlines several techniques that can be used to calculate limits, including direct substitution, factoring, rationalization, and limits involving infinity and trigonometric, exponential and two-sided limits.

Mathematical induction and divisibility rules

A complete and enhanced presentation on mathematical induction and divisibility rules with out any calculation.
Here are some defined formulas and techniques to find the divisibility of numbers.

CMSC 56 | Lecture 11: Mathematical Induction

Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas

Metric space

This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.

Recursion DM

The document discusses recursion and recursively defined sequences, functions, and algorithms. It provides examples of recursively defined sequences like powers of 2 and the Fibonacci sequence. Recursively defined functions are given, such as factorials and Fibonacci numbers. Recursive algorithms solve problems by reducing them to smaller instances of the same problem, like the Euclidean algorithm to find the greatest common divisor or a recursive Fibonacci algorithm. While recursive solutions are often shorter and more elegant, iterative algorithms using loops are usually more efficient.

Binomial Theorem

The binomial theorem provides a formula for expanding binomial expressions of the form (a + b)^n. It states that the terms of the expansion are determined by binomial coefficients. Pascal's triangle is a mathematical arrangement that shows the binomial coefficients and can be used to determine the coefficients in a binomial expansion. The proof of the binomial theorem uses mathematical induction to show that the formula holds true for any positive integer value of n.

Integral calculus

1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a

Number Theory - Lesson 1 - Introduction to Number Theory

This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes

Limit of functions

1) One-sided limits describe the value a function approaches as the input gets closer to a number from the left or right.
2) The limit of a function exists if and only if the one-sided limits are equal as the input approaches the number.
3) Limits at infinity describe the value a function approaches as the input increases without bound toward positive or negative infinity.

21 monotone sequences x

The document defines and discusses monotone sequences. A sequence {an} is defined as increasing if an < an+1, non-decreasing if an ≤ an+1, decreasing if an > an+1, and non-increasing if an ≥ an+1. Methods for determining if a sequence is monotone include the difference method, ratio method, and derivative method. Bounded and eventually monotone sequences are shown to converge according to the monotone sequence convergence theorem.

Number theory

This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.

Radical expressions

The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.

Relations in Discrete Math

A relation is a set of ordered pairs that shows a relationship between elements of two sets. An ordered pair connects an element from one set to an element of another set. The domain of a relation is the set of first elements of each ordered pair, while the range is the set of second elements. Relations can be represented visually using arrow diagrams or directed graphs to show the connections between elements of different sets defined by the relation.

number theory.ppt

This document discusses key concepts in number theory including divisibility, greatest common divisors, least common multiples, prime and composite numbers, relative primality, modular arithmetic, factorials, and applications. It defines these terms and provides examples. Greatest common divisors are the largest integers that divide two numbers. Least common multiples are the smallest integers divisible by two numbers. Prime numbers have only two factors and composite numbers are multiples of primes. Relative primality means two numbers have no common prime factors. Modular arithmetic uses the remainder of a division. Factorials are the product of integers up to a given number. Applications include cryptography.

Recurrence relations

This document discusses recurrence relations and their use in defining sequences. It introduces key concepts like recurrence relations, initial conditions, explicit formulas, and solving recurrence relations using techniques like backtracking or finding the characteristic equation. As examples, it examines the Fibonacci sequence and linear homogeneous recurrence relations of varying degrees.

Permutation and combination

1) The document discusses permutations and combinations, which are ways of arranging or selecting items from a group.
2) A permutation is an arrangement of items that considers order, while a combination disregards order.
3) Formulas are provided to calculate the number of permutations and combinations for a given number of items selected from a larger set.

Integration by partial fraction

this presentation covers the topic of integration by partial fraction...let me know if u finfd this helpful

Math induction principle (slides)

The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.

10-Sequences and summation.pptx

The document discusses sequences and summations. It provides examples and definitions of different types of sequences such as arithmetic and geometric sequences. It also discusses recurrence relations, which express a term in a sequence based on prior terms. Examples are provided to demonstrate finding terms of sequences given a recurrence relation. The document is a lecture on discrete mathematics concepts related to sequences.

Complex numbers And Quadratic Equations

This ppt is on the above mentioned topic with clear and interactive explainations and some examples too!

Remainder theorem

The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.

Limits

The document discusses the concept of limits. It explains that as the number of sides of a polygon increases, the area of the polygon approximates the area of the circle it is inscribed in, and the limit of the polygon's area is equal to the area of the circle. It also examines the limit of a function as x approaches 2 from both sides, and defines some fundamental rules of limits, such as the constant rule, sum rule, and multiplication rule. Finally, it outlines several techniques that can be used to calculate limits, including direct substitution, factoring, rationalization, and limits involving infinity and trigonometric, exponential and two-sided limits.

CMSC 56 | Lecture 11: Mathematical Induction

CMSC 56 | Lecture 11: Mathematical Induction

Metric space

Metric space

Recursion DM

Recursion DM

Binomial Theorem

Binomial Theorem

Integral calculus

Integral calculus

Number Theory - Lesson 1 - Introduction to Number Theory

Number Theory - Lesson 1 - Introduction to Number Theory

Limit of functions

Limit of functions

21 monotone sequences x

21 monotone sequences x

Number theory

Number theory

Radical expressions

Radical expressions

Relations in Discrete Math

Relations in Discrete Math

number theory.ppt

number theory.ppt

Recurrence relations

Recurrence relations

Permutation and combination

Permutation and combination

Integration by partial fraction

Integration by partial fraction

Math induction principle (slides)

Math induction principle (slides)

10-Sequences and summation.pptx

10-Sequences and summation.pptx

Complex numbers And Quadratic Equations

Complex numbers And Quadratic Equations

Remainder theorem

Remainder theorem

Limits

Limits

Mathematical induction and divisibility rules

A complete and enhanced presentation on mathematical induction and divisibility rules with out any calculation.
Here are some defined formulas and techniques to find the divisibility of numbers.

5.1 Induction

This document introduces mathematical induction. It defines the principle of mathematical induction as having two steps: (1) the basis step, which shows a statement P(1) is true, and (2) the inductive step, which assumes P(k) is true and shows P(k+1) is also true. It provides an example of climbing an infinite ladder to illustrate these steps. It also notes some important points about mathematical induction, such as that it is expressing a rule of inference and in proofs we show P(k) implies P(k+1) rather than assuming P(k) is true for all k.

X2 t08 02 induction (2013)

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.

Atomic theory notes

This document provides an overview of the development of atomic theory from ancient Greek philosophers to modern atomic structure. It summarizes key contributors and discoveries:
- Democritus proposed atoms as indivisible particles (5th century BC). John Dalton's atomic theory (1803) stated all matter is made of atoms that cannot be created, destroyed, or divided.
- J.J. Thomson's discovery of the electron (1897) showed atoms can be divided. Ernest Rutherford's gold foil experiment (1909) demonstrated atoms are mostly empty space with a dense nucleus.
- Niels Bohr's model (1913) showed electrons orbiting the nucleus in defined energy levels. Modern atomic theory describes electrons in

Fskik 1 nota

This document discusses mathematical induction as a method of proof. It explains that induction has three parts: the base case, the inductive hypothesis, and the inductive step. The base case shows that the statement holds true for the first relevant element. The inductive hypothesis assumes the statement holds true for an arbitrary element k. The inductive step then shows that if the statement holds true for k, it must also hold true for k+1. The document provides examples of proofs by induction, such as showing the sum of the first n odd integers is n^2. It emphasizes that induction only proves statements, it does not generate answers. The key idea is to manipulate the inductive step so it can substitute part of the induct

Predicates and quantifiers

This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can represent concepts like "all parking spaces are full" or "some parking space is full." Laws of quantifier equivalence and negation rules with quantifiers are also presented.

Divisibility Rules

Divisibility refers to whether a number can be divided by another number without a remainder. A number is divisible by another number if when you divide them, the result is a whole number. The document then provides rules for determining if a number is divisible by 2, 3, 5, 6, 8, 9, 10, and 4. It explains that you cannot divide by 0 because there is no number that when multiplied by 0 equals the original number.

Mathematics - Divisibility Rules From 0 To 12

This document discusses divisibility rules for numbers 0-12. It provides shortcuts to determine if a number is divisible by certain divisors without performing long division. For each rule, it explains the pattern to look for in the digits of the number. For example, a number is divisible by 2 if its last digit is even, by 5 if its last digit is 0 or 5, and by 11 if adding all even and odd digits separately and subtracting the results equals 0. The document aims to teach efficient ways to test divisibility through brief explanations and examples.

Mgc bohr

Niels Bohr developed his atomic theory in 1913 which proposed that electrons orbit the nucleus in fixed, quantized energy levels. The Bohr model could reproduce the emission spectrum of hydrogen and predict wavelengths of photons emitted during transitions between energy levels. While an oversimplification, the Bohr model was pioneering in introducing quantum mechanics and helped establish the modern understanding of atomic structure.

Binomial theorem

The Binomial Theorem provides a formula for expanding binomial expressions of the form (a + b)^n. It states that the terms follow a predictable pattern based on exponents of the variables a and b and coefficients determined by Pascal's Triangle. The theorem was first discovered by Isaac Newton and can be written as a general formula involving factorials and binomial coefficients. It allows for the easy expansion of binomials without having to manually multiply out each term.

Predicates and Quantifiers

This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.

THE BINOMIAL THEOREM

THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)

Mathematical induction

This document appears to contain student information, listing a name "Ankur Amba", class "XI-C", and a number "32", which is likely a roll or identification number. In 3 lines of text, the document provides basic details about a student.

C++ programs

The document contains code snippets and descriptions for various C++ programs, including:
1) An abstract class example with Shape as the base class and Rectangle and Triangle as derived classes, demonstrating polymorphism.
2) A program that counts the words in a text by getting user input and parsing for whitespace.
3) An Armstrong number checker that determines if a number is an Armstrong number based on the sum of its digits.
4) Various other examples like binary search, complex number arithmetic, stacks, inheritance, and converting between Celsius and Fahrenheit temperatures.

Mathematical induction and divisibility rules

Mathematical induction and divisibility rules

Proof

Proof

5.1 Induction

5.1 Induction

X2 t08 02 induction (2013)

X2 t08 02 induction (2013)

Atomic theory notes

Atomic theory notes

Fskik 1 nota

Fskik 1 nota

Predicates and quantifiers

Predicates and quantifiers

Divisibility Rules

Divisibility Rules

Mathematics - Divisibility Rules From 0 To 12

Mathematics - Divisibility Rules From 0 To 12

Mgc bohr

Mgc bohr

Binomial theorem

Binomial theorem

Predicates and Quantifiers

Predicates and Quantifiers

THE BINOMIAL THEOREM

THE BINOMIAL THEOREM

Mathematical induction

Mathematical induction

C++ programs

C++ programs

1.Prove 3 n^3 + 5n , n = 1For n = 1, 1^3 + 51 = 1 + 5 = 6, a.pdf

1.
Prove 3 | n^3 + 5n , n >= 1
For n = 1, 1^3 + 5*1 = 1 + 5 = 6, and 6/3 = 2, so 3 | n^3 + 5n
Assume for n = k
3 | k^3 + 5k
Then, for k+1
(k+1)^3 + 5(k+1) = k^3 + 3k^2 + 3k + 1 + 5k + 5 = k^3 + 5k + (3k^2 + 3k + 6) = k^3 + 5k +
3(k^2 + k + 2)
From the assumption, 3 | k^3 + 5k, or k^3 + 5k = 3z for some integer z.
Thus, k^3 + 5k + 3(k^2 + k + 2) = 3z + 3(k^2 + k + 2) = 3(z + k^2 + k + 2)
Thus, 3 | n^3 + 5n for n = k+1, and the proof by induction is complete.
2. Show 2 + 5 + ...(3n - 1) = n(3n+1)/2
For n = 1, the left hand side is 2 and the right hand side is n(3n+1)/2 = 1(3*1+1)/2 = 1(4)/2 = 2
2 = 2
Assume for n = k
2 + 5 + ...(3k - 1) = k(3k+1)/2
for n = k+1
2 + 5 + ...+ (3k - 1) + (3(k+1) - 1) =
(2 + 5 + ...+ (3k - 1) ) + 3k+2 = from assumption
k(3k+1)/2 + 3k + 2 =
3k^2/2 + k/2 + 3k + 2 =
3k^2/2 + 7/2 k + 2 =
(3k^2 + 7k + 4)/2 =
(k+1)(3k+4)/2 =
(k+1)(3k+3 + 1)/2 =
(k+1)(3(k+1) + 1)/2 =
n(3n+1)/2 for n = k+1
Thus, we have proven for n=k+1 and our proof is complete.
3. Show for n >= 1, 4 | 5^n - 1
For n = 1, 5^1 - 1 = 5 - 1 = 4 and 4 | 4
Assume for n = k
4 | 5^k - 1, or 5^k - 1 = 4z for some integer z.
Then, for n = k+1
5^(k+1) - 1 = 5^(k+1) - 1 - 5^k + 5^k = 5^k - 1 + 5^(k+1) - 5^k = 5^k - 1 + 5*5^k - 5^k = 5^k - 1
+ (5-1)*5^k =
5^k - 1 + 4*5^k = from assumption, 4z + 4*5^k = 4(z + 5^k)
Thus, 4 | 5^(k+1) - 1 or 4|5^n - 1 for n = k+1, and our proof by induction is complete.
4. Show for n >= 1 7|9^n - 2^n
For n = 1 9^1 - 2^1 = 9 - 2 = 7 and 7 | 7
Assume for n = k, 7|9^k - 2^k, or 7z = 9^k - 2^k
Then, for n = k+1,
9^(k+1) - 2^(k+1) = 9*9^k - 2 *2^k = (7+2)*9^k - 2 * 2^k = 7 * 9^k + 2 * 9^k - 2 * 2^k =
7 * 9 ^ k + 2 * (9^k - 2^k) = from the assumption,
7 * 9 ^ k + 2 * 7z = 7( 9 ^ k + 14z)
Thus, 7 | 9^(k+1) - 2^(k+1), or 9^n - 2^n for n = k+1 and the proof by induction is complete.
5. 1/2 + ... + n/2^n = 2 - (n+2)/2^n
For n = 1, we have 1/2 on the left
On the right, we have 2 - (1+2)/2^1 = 2 - 3/2 = 1/2
1/2 = 1/2
Assume for n = k 1/2 + ... + k/2^k = 2 - (k+2)/2^k
Then, for n = k + 1
1/2 + ... + k/2^k + (k+1)/2^(k+1) =
(1/2 + ... + k/2^k) + (k+1)/2^(k+1) = from assumption
2 - (k+2)/2^k + (k+1)/2^(k+1) =
2 - 2(k+2)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4)/2^(k+1) + (k+1)/2^(k+1) =
2 - (2k+4 -(k+1))/2^(k+1) =
2 - (2k+4 - k - 1)/2^(k+1) =
2 - (k+3)/2^(k+1) =
2 - ((k+1)+2)/2^(k+1) =
2 - (n+2)/2^n for n = k+1 and the proof is complete.
Show 2^n + n <= 3^n for n >= 1 (or n >= 0
Note: for n = 0, 2^0 + 0 = 1 + 0 = 1, and 3^0 = 1, and 1 <= 1
for n = 1, 2^1 + 1 = 2 + 1 = 3, and 3^1 = 3, and 3 <= 3
Then, assume for n = k
2^k + k <= 3^k
Then, for n = k+ 1
2^(k+1) + (k+1) =
2*2^k + (k+1) =
2^k + k + 2^k + 1 <= by assumption
3^k + 2^k + 1 <= by assumption, as 1 <= k
3^k + 3^k = 2 * 3^k < as 3^k > 0
2 * 3^k + 3^k = (2+1)*3^k = 3*3^k = 3^(k+1)
Thus, 2^(k+1) + (k+1) < 3^(k+1) for k >= 1 or
2^(n) + (n) < 3^(n) for n >= 2, and
2^(n) + (n) <= 3^(n) for all natural n
B. Extended induction.
1. This problem actually does not require extended induction but can use regular ind.

Induction q

The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being

Induction q

The document proves mathematical statements using induction. It shows:
1) Induction can be used to prove the summation formula 1 + 2 + 3 + ... + n = n(n+1)/2. It shows the base case of n=1 is true, and assumes the formula is true for n=k to prove it is true for n=k+1.
2) Induction can prove inequalities, like n < 2n for all positive integers n. It shows the base case is true and assumes the statement is true for n=k to prove it is true for n=k+1.
3) Induction can prove divisibility properties, like n3 - n being

5.4 mathematical induction t

The document outlines the steps of mathematical induction:
1. Declare that an induction argument will be used
2. Verify the statement is true for the base case (usually n=1)
3. Assume the statement is true for some integer n=k
4. Use the induction assumption to prove the statement is true for n=k+1
It then provides examples demonstrating how to use induction to prove statements like sums and inequalities for all natural numbers.

Text s1 21

The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.

Activ.aprendiz. 3a y 3b

The document provides solutions to exercises on mathematical induction. It presents proofs for 7 formulas involving sums and sequences. Each proof follows the same structure: show the base case is true, assume the statement is true for some integer k, and prove it is true for k+1 using algebraic manipulations. This establishes the formula is true for all positive integers n by the principle of mathematical induction.

11-Induction CIIT.pptx

This document discusses mathematical induction. It defines induction as generalizing statements from facts and provides two steps to prove a statement P(n) is true for all natural numbers n: 1) the basis step verifies P(1) is true, and 2) the inductive step shows P(k) implies P(k+1) is true for all positive integers k. Several examples are provided to illustrate these steps, such as proving formulas for the sum of the first n positive integers and odd integers, and the sum of terms in a geometric progression.

1113 ch 11 day 13

The document discusses using mathematical induction to prove the formula:
3 + 5 + 7 +...+ (2k + 1) = k(k + 2)
It provides the base case of p(1) and shows that it is true. It then assumes p(k) is true, and shows that p(k+1) follows by algebraic manipulations. This completes the induction proof.

Please provide a detailed explanation. Prove by induction on n that .pdf

Please provide a detailed explanation. Prove by induction on n that nk =2K/(2K - 1)2(2K + 1)2 =
n2 + n/(2n + 1)2.|
Solution
Consider for n=1
LHS=2*1/[(2-1)^2*(2+1)^2]
=2/9
RHS=(1^2+1)/(2+1)^2
=2/9
Hence the given expression is true for n=1
consider the given expression is true for n
hence
k=1n (2k/(2k-1)^2(2k+1)^2)=n^2+n/(2n+1)^2---------->A
consider for n+1
LHS= k=1n+1 (2k/(2k-1)^2(2k+1)^2)
=[k=1n (2k/(2k-1)^2(2k+1)^2) ]+(2*(n+1)/(2(n+1)-1)^2*(2(n+1)+1)^2)
=(n^2+n)/(2n+1)^2 +(2(n+1)/(2n+1)^2*(2n+3)^2)
=[(n+1)/(2n+1)^2]*(n+(2/(2n+3)^2))
=[(n+1)/(2n+1)^2]*(n*((2n+1)+2)^2)+2)/(2n+3)^2
=[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+n*4+2)/(2n+3)^2
=[(n+1)/(2n+1)^2]*(n*(2n+1)^2+n*2*(2n+1)*2+2*(2n+1))/(2n+3)^2
=[(n+1)/(2n+1)]*(n*(2n+1)+4n+2)/(2n+3)^2
=[(n+1)/(2n+1)]*(n*(2n+1)+2*(2n+1))/(2n+3)^2
=(n+1)*(n+2)/(2n+3)^2
=(n+1)(n+2)/(2n+3)^2
RHS=((n+1)^2+(n+1))/(2(n+1)+1)^2
=(n+1)(n+1+1)/(2n+3)^2
=(n+1)(n+2)/(2n+3)^2
since LHS=RHS
The given expression is true for n+1
Hence by principle of mathematical induction,the given expression is true for all values of n..

Further mathematics notes zimsec cambridge zimbabwe

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Slide subtopic 5

The document discusses mathematical induction. It begins by defining the principle of induction and stating that it has two parts: the basic step and inductive step. It then gives an example proof using induction. Specifically, it proves that the sum of n times the nth factorial from 1 to n equals (n+1)! - 1. It shows the base case of n=1 holds and assumes the inductive hypothesis is true for k. Then it uses this to prove the statement is true for k+1, completing the induction proof.

第二次作業

1. The document proves through mathematical induction that the formula nj=1 j^2 = n(n+1)(2n+1)/6 for the sum of squares from 1 to n is true for all positive integers n.
2. It also proves that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces that each cover three squares, again using mathematical induction.
3. Mathematical induction is used to prove both formulas by showing the base case holds and assuming the formula is true for an integer n to prove it is true for n+1.

Maths04

The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.

Succesive differntiation

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The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.

Assign1 21314-sol

This document contains solutions to 5 calculus assignment problems:
1) Finding values of x such that an inequality is satisfied, with the solution being 3 < x < 4.
2) Proving that the square of any odd integer is odd.
3) Proving there is no rational number whose square is 2.
4) Determining the value of x where a quadratic equation has a minimum, which is x = 1.
5) Proving an equation involving sums and powers using mathematical induction, for all natural numbers n and where r ≠ 1.

New Pythagorean Triples copy

Pythagorean triples are whole number sets that satisfy the Pythagorean theorem, where a2 + b2 = c2. The document discusses properties of Pythagorean triples and how they relate to rational points on the unit circle. It presents theorems showing that every basic Pythagorean triple corresponds to a rational point on the unit circle, and vice versa. Formulas are derived for generating Pythagorean triples from a given rational slope of a line passing through (-1,0) and a point on the unit circle. The document also briefly discusses 60-degree triangles, whose sides satisfy the equation c2 = a2 + b2 - ab, relating to an ellipse rather than a circle.

Maths05

The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.

A proof induction has two standard parts. The first establishing tha.pdf

A proof induction has two standard parts. The first establishing that a theorem is true for some
small (usually almost always trivial. Next, an inductive hypothesis is the theorem is assumed to
be true for all cases up to some the theorem is then shown to be true for the next value proves
the theorem (as long as k is finite). As an example, we prove that the Fibonacci numbers, F_4 =
5, ..., F_i = F_i-1 + F_i-2, satisfy F_i
Solution
The induction proof works by verifying whether the example works for all possible values.. Let
us see how it works..
For example :: Consider E(n) 12 + 22 + 32 + ... + n2 = (1/6). n.(n+1).(2n+1)
Now, we need to prove that the above example is true for all possible values..
So, first we consider n = 1.. For n = 1, obviously the sequence is true..
For n=1, LHS = 12
RHS = (1/6).1.(1+1).(2+1) = 6/6 = 1
here LHS = RHS..
Now we have to prove that : if E(k) is true, then E(k+1) is true..
Let us consider E(K) is true..
We have to prove that property is valid for n = k+1
We have to prove: 12 + 22 + 32 + ... + k2 + (k+1)2 = (1/6).(k+1).(k+2)(2k+3)
Left side
= 12 + 22 + 32 + ... + k2 + (k+1)2
= (12 + 22 + 32 + ... + k2) + (k+1)(k+1)
= (1/6). k.(k+1).(2k+1) + (k+1)(k+1)
= (1/6).(k+1). [ k(2k+1) + 6(k+1)]
= (1/6).(k+1).(2 k2 + 7k + 6)
Right side
= (1/6).(k+1).(k+2)(2k+3)
= (1/6).(k+1).(2k2 + 3k + 4k + 6)
Here, LHS = RHS.. I.e., this is true for E(k+1) therefore our assumption is true..
That means the equation is true for all possible values by principle of induction...

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1.Prove 3 n^3 + 5n , n = 1For n = 1, 1^3 + 51 = 1 + 5 = 6, a.pdf

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Induction q

Induction q

Induction q

Induction q

5.4 mathematical induction t

5.4 mathematical induction t

Text s1 21

Text s1 21

Activ.aprendiz. 3a y 3b

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1113 ch 11 day 13

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Please provide a detailed explanation. Prove by induction on n that .pdf

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Further mathematics notes zimsec cambridge zimbabwe

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Slide subtopic 5

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第二次作業

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You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
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Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP

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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.

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- 2. 1 + 3 + 5 + ... + (2n-1) = n2 1. Show it is true for n=1 1 = 12 is True 2. Assume it is true for n=k 1 + 3 + 5 + ... + (2k-1) = k2 is True Now, prove it is true for "k+1“ 1+3+5+...+(2k-1)+ 2(k+1)-1)=(k+1)2 ... ? We know that 1 + 3 + 5 + ... + (2k-1) = k2 (the assumption above), so we can do a replacement for all but the last term: k2 + (2(k+1)-1) = (k+1)2
- 3. Now expand all terms: k2 + 2k + 2 - 1 = k2 + 2k+1 And simplify: k2 + 2k + 1 = k2 + 2k + 1 They are the same! So it is true. So: 1 + 3 + 5 + ... + (2(k+1)- 1) = (k+1)2 is True
- 4. 1 + 2 + 3 + ... + n = n(n + 1) / 2 let S(n) be the statement 1+2+3+... n=n(n+1)/2 S(1) is the statement 1=1(1+1/2. Thus S(1) is true. We suppose that S(k) is true and prove that S(k + 1) is true. Thus, we assume that 1 + 2 + 3 + ... + k = k(k + 1) / 2 and prove that 1+2+3+...+k+k+1=(k+1)(k+1+1) / 2
- 5. If we add k + 1 to both sides of the equality in S(k), then on the left side of the sum, we obtain the left side of equality in S(k + 1). Our hope is that the right of the sum equals the right side of S(k + 1). Let us check: Adding K + 1 to both sides of S(k) we get: 1 + 2 + 3 + ... + k + k + 1 = k(k + 1) / 2 + (k + 1) = (k + 1)(k / 2 + 1) = (k + 1)(k / 2 + 2 / 2) = (k + 1)(k + 2) / 2 = (k + 1)(k + 1 + 1) / 2 Hence, if S(k) is true, then S(k + 1) is true. Therefore, 1 + 2 + 3 + ... + n = n(n + 1) / 2 for each positive integer n.
- 6. 1 + 3 + 5 + . . . + (2n − 1) = n2 for any integer n ≥ 1. Proof: STEP 1: For n=1 (1.2) is true, since 1 = 12 STEP 2: Suppose (1.2) is true for some n = k ≥ 1, that is 1 + 3 + 5 + . . . + (2k − 1) = k 2
- 7. STEP 3: Prove that (1.2) is true for n=k+1, that is 1+3+5+...+(2k−1)+(2k+1)?=(k+) 2 We have: 1+3+5+...+(2k−1)+(2k+1)=k 2+(2k+1)= (k + 1) 2
- 8. Sn = 1 + 3 + 5 + 7 + . . . + (2n-1) = n2 First, we must show that the formula works for n = 1. 1. For n = 1 S1 = 1 = 12 The second part of mathematical induction has two steps. The first step is to assume that the formula is valid for some integer k. The second step is to use this assumption to prove that the formula is valid for the next integer, k + 1. 2. Assume Sk = 1 + 3 + 5 + 7 + . . . + (2k-1) = k2 is true, show that Sk+1 = (k + 1)2 is true.
- 9. Sk+1 = 1+3+5+7. . .+(2k–1)+[2(k+1)–1] = [1+3+5+7+. . .+(2k–1)]+(2k+2–1) = Sk+(2k+1) = k2+2k+1 = (k+1)2
- 10. 12 + 22 + ... + n2 = n( n + 1 )( 2n + 1 )/6 If n = 0, then LHS = 02 = 0, and RHS = 0 * (0 + 1)(2*0 + 1)/6 = 0 Hence LHS = RHS. 12 + 22 + ... + n2 = n( n + 1 )( 2n + 1 )/6 --- Induction Hypothesis To prove this for n+1, first try to express LHS for n+1 in terms of LHS for n, and use the induction hypothesis. Here let us try LHS for n + 1 = 12 + 22 + ... + n2 + (n + 1)2 = ( 12 + 22 + ... + n2 ) + (n + 1)2
- 11. Using the induction hypothesis, the last expression can be rewritten as n( n + 1 )( 2n + 1 )/6 + (n + 1)2 Factoring (n + 1)/6 out, we get ( n + 1 )( n( 2n + 1 ) + 6 ( n + 1 ) )/6 = ( n + 1 )( 2n2 + 7n + 6 )/6 = ( n + 1 )( n + 2 )( 2n + 3 )/6 , which is equal to the RHS for n+1. Thus LHS = RHS for n+1.
- 12. n , n3 + 2n is divisible by 3. If n = 0, then n3 + 2n = 03 + 2*0 = 0. So it is divisible by 3. Induction: Assume that for an arbitrary natural number n, n3 + 2n is divisible by 3. --- Induction Hypothesis To prove this for n+1, first try to express (n+1)3 + 2( n + 1 ) in terms of n3 + 2n and use the induction hypothesis.
- 13. (n+1)3+2(n+1)=(n3+3n2+3n+1)+( 2n + 2 ) = ( n3 + 2n ) + ( 3n2 + 3n + 3 ) = ( n3 + 2n ) + 3( n2 + n + 1 ) which is divisible by 3, because ( n3 + 2n ) is divisible by 3 by the induction hypothesis.
- 14. 2 + 4 + ... + 2n = n( n + 1 ) If n=0, then LHS=0, and RHS=0*(0+1)=0 Hence LHS = RHS Assume that for an arbitrary natural number n, 0+2+...+2n=n(n+1) --- Induction Hypothesis To prove this for n+1, first try to express LHS for n+1 in terms of LHS for n, and somehow use the induction hypothesis.
- 15. Here let us try LHS for n+1=0+2+...+2n+2(n+1)=(0+2+ ...+2n)+2(n+1) Using the induction hypothesis, the last expression can be rewritten as n( n + 1 ) + 2(n + 1) Factoring (n + 1) out, we get (n + 1)(n + 2) , which is equal to the RHS for n+1. Thus LHS = RHS for n+1.