Explanation Concept used: According to the “Principle of Mathematical Induction”, to prove that propositional function, P ( n ) is true for all positive integers n , we have to perform the following two steps: BASIS STEP: We verify that P ( a ) is true, where a is a particular integer like 1 , 2 , 3 … etc INDUCTIVE STEP: We show that the conditional statement P ( k ) → P ( k + 1 ) is true for all positive integers k > a . Proof: let P ( n ) , be the proposition that for some non-negative integer n , n 2 ≤ n ! , is true BASIS STEP: Since n is a non-negative integer, and we don’t know value of a we verify that P ( n ) , for various value like 1 , 2 , 3 … etc. Plug n = 1 , in above propositional function, we get 1 2 ≤ 1! 1 ≤ 1 , which is true so we can set a = 1 Accordingly we can also change our proposition P ( n ) . P ( n ) , is the proposition that If n is an integer such that n ≥ 1 , then, n 2 ≤ n ! , is true Now, since we have already verified the basis step P ( 1 ) , for this proposition, we now perform the Inductive step for this proposition. INDUCTIVE STEP: We start by assuming P ( n ) , is true for some positive integer k , i.e. we are assuming P ( k ) is true. This assumption is called “Inductive Hypothesis”. Now using this “Inductive Hypothesis”, we need to show that P ( k + 1 ) is also true. Assuming P ( k ) is true, we must have k 2 ≤ k ! for some positive integer k ≥ 1 Next, for P ( k + 1 ) , we need to show ( k + 1 ) 2 ≤ ( k + 1 ) ! , for some k such that ( k + 1 ) ≥ 1 Left hand side is, ( k + 1 ) 2 = k 2 + 2 k + 1 , now since k ≥ 1 , replacing 1 by k (</

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 5.1, Problem 22E

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For which non-negative integer n, n2 ≤ n! is valid and prove the same.

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Chapter 5.1, Problem 21E

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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

Ch. 5.1 - re are infinite]y many stations on a train route....Ch. 5.1 - pose that you know that a golfer plays theho1e of...Ch. 5.1 - P(n) be the statement...Ch. 5.1 - P(n) be the statementthat 13+ 23+ ... + n3=...Ch. 5.1 - ve...Ch. 5.1 - ve that1.1!+2.2!+...n.n!=(n+1)!1whenevernis a...Ch. 5.1 - ve that3+3.5+3.52+...+3.5n=3(5n+11)/4whenevernis a...Ch. 5.1 - ve that22.7+2.72...+2(7)n=(1(7)n+1)/4whenevernis a...Ch. 5.1 - a)Find a formula for the sum of the firstneven...Ch. 5.1 - a) Find a formula for 112+123++1m(n+1) by...

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For which non-negative integer n , n 2 ≤ n ! is valid and prove the same.

Solution Summary: The author explains how to prove that propositional function, P(n) is true for all positive integers n.

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For which non-negative integer n, n2 ≤ n! is valid and prove the same.

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For which non-negative integer n , n 2 ≤ n ! is valid and prove the same.

Solution Summary: The author explains how to prove that propositional function, P(n) is true for all positive integers n.

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To find: For which non-negative integer n, n2 ≤ n! is valid and prove the same.

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Chapter 5 Solutions

To find:

For which non-negative integer n, n2 ≤ n! is valid and prove the same.