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(5.3.1) Mathematical Induction: Proving Money Compositions with 3¢ and 8¢ Coins
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This video is dedicated to demonstrating how mathematical induction can be applied to prove that any amount of money, starting from 14 cents, can be composed using just 3¢ and 8¢ coins. Following the structure of proposition 5.3.1 from our textbook, which showed that any amount of 8 cents and above could be obtained using 3¢ and 5¢ coins, we embark on a similar journey with a new set of coins. We start our proof by defining the proposition P(n), which states that 'n' cents can be achieved using 3¢ and 8¢ coins, specifically when 'n' is an integer greater than or equal to 14.
The proof kicks off with the base case, where I demonstrate that the smallest integer satisfying our conditions, 14 cents, can indeed be composed using two 3¢ coins and one 8¢ coin, establishing the truth of P(14). This initial step is crucial as it sets the foundation for our induction process.
Moving on to the inductive step, the goal is to show that if P(k) is true for any integer k greater than or equal to 14, then P(k+1) will also hold true. Assuming P(k) is valid, we explore how to transition from 'k' cents to 'k+1' cents using our available coins. The proof utilizes strategic exchanges of coins to increment the total value by exactly one cent, whether by swapping an 8¢ coin for three 3¢ coins or adjusting the composition of coins when only 3¢ coins are used.
By breaking down the proof into these two scenarios, we ensure that for any amount k (where k≥14), there is a method to construct 'k+1' cents, thereby proving the proposition by induction. This approach not only solidifies our understanding of mathematical induction but also highlights its practical application in solving real-world problems.
Tags: #MathematicalInduction, #ProofTechniques, #PropositionProof, #CoinComposition, #MathEducation, #ProblemSolving, #InductiveProof, #MathTutorial, #MathConcepts, #LearningMath, #Mathematics, #MathHelp, #StudyMath, #EducationalContent, #MathLessons, #MathSkills, #Tutorial, #MathVideos, #TeachingMath, #Algebra, #DiscreteMath, #NumberTheory, #Logic, #CriticalThinking
The proof kicks off with the base case, where I demonstrate that the smallest integer satisfying our conditions, 14 cents, can indeed be composed using two 3¢ coins and one 8¢ coin, establishing the truth of P(14). This initial step is crucial as it sets the foundation for our induction process.
Moving on to the inductive step, the goal is to show that if P(k) is true for any integer k greater than or equal to 14, then P(k+1) will also hold true. Assuming P(k) is valid, we explore how to transition from 'k' cents to 'k+1' cents using our available coins. The proof utilizes strategic exchanges of coins to increment the total value by exactly one cent, whether by swapping an 8¢ coin for three 3¢ coins or adjusting the composition of coins when only 3¢ coins are used.
By breaking down the proof into these two scenarios, we ensure that for any amount k (where k≥14), there is a method to construct 'k+1' cents, thereby proving the proposition by induction. This approach not only solidifies our understanding of mathematical induction but also highlights its practical application in solving real-world problems.
Tags: #MathematicalInduction, #ProofTechniques, #PropositionProof, #CoinComposition, #MathEducation, #ProblemSolving, #InductiveProof, #MathTutorial, #MathConcepts, #LearningMath, #Mathematics, #MathHelp, #StudyMath, #EducationalContent, #MathLessons, #MathSkills, #Tutorial, #MathVideos, #TeachingMath, #Algebra, #DiscreteMath, #NumberTheory, #Logic, #CriticalThinking
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