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Induction:

Induction:

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter13: Sequences And Series
Section13.5: Mathematical Induction
Problem 30E
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please help with these problems, second photo is practice problem close to these problems, thank you 

Problem 1 Induction: Factorial Inequality Using induction, show that n! < n" for all n > 1. (You may find it helpful to see the practice problem on induction to see what is required of your proof). Solution: We use induction on a sequence of statements: • 2! < 22 (statement 2) 3! < 33 (statement 3) n! < n" (statement n) Base Case (statement 2): 2! = 2 * 1 = 2 < 4 = 22 so that 2! < 22 Inductive Step: Assume statement n is true, that is n! < n". Then: (n + 1)! = n! * (n + 1) < n" + (n + 1) < (n + 1)" * (n+ 1) = (n + 1)"+1 Where we use statement n in the second line and the fact that n" < (n +1)" in the third (since n is positive). This statement shows that (n + 1)! < (n + 1)n+1. By induction, n! < n" for all n > 1.
Transcribed Image Text:Problem 1 Induction: Factorial Inequality Using induction, show that n! < n" for all n > 1. (You may find it helpful to see the practice problem on induction to see what is required of your proof). Solution: We use induction on a sequence of statements: • 2! < 22 (statement 2) 3! < 33 (statement 3) n! < n" (statement n) Base Case (statement 2): 2! = 2 * 1 = 2 < 4 = 22 so that 2! < 22 Inductive Step: Assume statement n is true, that is n! < n". Then: (n + 1)! = n! * (n + 1) < n" + (n + 1) < (n + 1)" * (n+ 1) = (n + 1)"+1 Where we use statement n in the second line and the fact that n" < (n +1)" in the third (since n is positive). This statement shows that (n + 1)! < (n + 1)n+1. By induction, n! < n" for all n > 1.
Induction (Powerset Cardinality) Using induction, show that the cardinality of the powerset of A is 2" where n is the number of elements in A. In other words, given a set A with |A| = n then |P(A)| = 2". (You may find it helpful to see the practice problem on induction to see what is required of your proof). Induction (Geometric Series) Using induction, prove the geometric series formula: 1- r" i=1 (You may find it helpful to see the practice problem on induction to see what is required of your proof).
Transcribed Image Text:Induction (Powerset Cardinality) Using induction, show that the cardinality of the powerset of A is 2" where n is the number of elements in A. In other words, given a set A with |A| = n then |P(A)| = 2". (You may find it helpful to see the practice problem on induction to see what is required of your proof). Induction (Geometric Series) Using induction, prove the geometric series formula: 1- r" i=1 (You may find it helpful to see the practice problem on induction to see what is required of your proof).
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