3) Use the mathematical induction to show the inequalityn! > 10", when n ≥ no,for some fixed integer no > 0.

Let,,for some fixed integer Obviously Now let us assume that In other words, Now [Using Hypothesis…

Answered: (3) Use the mathematical induction to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3) Use the mathematical induction to show the inequality
n! > 10", when n ≥ no,
for some fixed integer no > 0.

Expert Solution

Step 1: Mathematical Induction use to prove inequality here

Let, $P open parentheses n close parentheses space b e space t h e space s t a t e m e n t space s u c h space t h a t space P open parentheses n close parentheses space colon space n factorial greater than 10 to the power of n comma w h e n space n greater or equal than n subscript 0$ ,for some fixed integer $n subscript 0$

Obviously $P open parentheses n close parentheses space i s space t r u e space f o r space n equals n subscript 0$

Now let us assume that $P left parenthesis m right parenthesis space i s space t r u e space comma m greater than n subscript 0$

In other words, $m factorial greater than 10 to the power of m$

Now $open parentheses m plus 1 close parentheses factorial equals open parentheses m plus 1 close parentheses m factorial greater than 10 space cross times 10 to the power of m equals 10 to the power of m plus 1 end exponent$ [Using Hypothesis ]

[ here $m plus 1 greater or equal than 10 space comma b e c a u s e space i f space n o t space t h e n space m less or equal than 8 rightwards double arrow n subscript 0 less than 8 rightwards double arrow 8 factorial greater than 10 to the power of 8 comma a space c o n t r a d i c t i o n$ ]

Thus $P open parentheses m plus 1 close parentheses space i s space t r u e space w h e n space P open parentheses m close parentheses space i s space t r u e$ .