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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/zz%3A_Back_Matter/22%3A_Appendix_B%3A_Mathematical_InductionThere is a variant of one of the bijections we used to prove the Pascal Equation that comes up in counting the subsets of a set. In the next problem, it will help us compute the total number of subsets...There is a variant of one of the bijections we used to prove the Pascal Equation that comes up in counting the subsets of a set. In the next problem, it will help us compute the total number of subsets of a set, regardless of their size. Our main goal in this problem, however, is to introduce some ideas that will lead us to one of the most powerful proof techniques in combinatorics (and many other branches of mathematics), the principle of mathematical induction.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/05%3A_Proof_Techniques_II_-_Induction/5.01%3A_The_Principle_of_Mathematical_InductionThe Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lift...The Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lifting yourself into the air. Despite the indisputable fact that proofs by PMI often feel like magic, we need to convince you of the validity of this proof technique. It is one of the most important tools in your mathematical kit!
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/07%3A_Sequences_and_Series_Mathematical_Induction_and_the_Binomial_Theorem/7.03%3A_Mathematical_InductionThis section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if th...This section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if the statement holds for one integer, it also holds for the next. Examples illustrate how to apply induction to verify formulas and inequalities, providing a foundation for proving mathematical statements systematically.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.6%3A_Mathematical_Induction_-_An_IntroductionConsider P(n):n2+n+11 is prime. In the inductive step, we want to prove that P(k)⇒P(k+1) for ANY k≥1. The following table verifies that it is tru...Consider P(n):n2+n+11 is prime. In the inductive step, we want to prove that P(k)⇒P(k+1) for ANY k≥1. The following table verifies that it is true for 1≤k≤9: n123456789n2+n+111317233141536783101 Nonetheless, when n=10, n2+n+11=121 is composite.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/09%3A_Sequences_and_the_Binomial_Theorem/9.03%3A_Mathematical_InductionHere we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated previously.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus_(2e)/05%3A_Sequences_Summations_and_Logic/5.05%3A_Mathematical_InductionThis section explains the principle of mathematical induction, a proof technique used to establish the truth of statements for all natural numbers. It covers the base step and inductive step, providin...This section explains the principle of mathematical induction, a proof technique used to establish the truth of statements for all natural numbers. It covers the base step and inductive step, providing a structured process to complete an induction proof. Examples demonstrate how to apply induction to sequences, inequalities, and mathematical identities.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/07%3A_Sequences_and_the_Binomial_Theorem/7.03%3A_Mathematical_InductionHere we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated previously.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.04%3A_Mathematical_Induction_-_An_IntroductionMathematical induction can be used to prove that an identity is valid for all integers n≥1 .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/02%3A__Induction_and_Recursion/2.01%3A_Some_Examples_of_Mathematical_IntroductionThe principle of mathematical induction states that in order to prove a statement about an integer n, if we can 1) Prove the statement when n = b, for some fixed integer b, and 2) Show that the tru...The principle of mathematical induction states that in order to prove a statement about an integer n, if we can 1) Prove the statement when n = b, for some fixed integer b, and 2) Show that the truth of the statement for n = k−1 implies the truth of the statement for n = k whenever k > b, then we can conclude the statement is true for all integers n ≥ b.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/09%3A_Sequences_and_the_Binomial_Theorem/9.03%3A_Mathematical_InductionHere we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated previously.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_(Stitz-Zeager)_-_Jen_Test_Copy/09%3A_Sequences_and_the_Binomial_Theorem/9.03%3A_Mathematical_InductionHere we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated previously.
