4)We know thatal%3Dand a!=1.2...q.19 i(9-0)Prove thut Yn7s, (2n) <2n-2

Given, ab=a!(a-b)!b! and a!=1.2....a

Answered: 4) We know that () and a!=1.2...a.…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.2: Matrix Algebra

Problem 21EQ: Prove the half of Theorem 3.3 (e) that was not proved in the text.

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4)
We know that
al
%3D
and a!=1.2...q.
19 i(9-0)
Prove thut Yn7s, (2n) <2n-2

Expert Solution

Step 1

Given,

$(\begin{matrix} a \\ b \end{matrix}) = \frac{a!}{(a - b)! b!}$ and a!=1.2....a

Step 2

Prove that for all $n \geq 5$ ,

$(\begin{matrix} 2 n \\ n \end{matrix}) < 2^{2 n - 2}$

Use mathematical induction method to prove the above condition:

n=5

$(\begin{matrix} 2 (5) \\ 5 \end{matrix}) < 2^{2 (5) - 2} (\begin{matrix} 10 \\ 5 \end{matrix}) < 2^{10 - 2} \frac{10!}{(10 - 5)! 5!} < 2^{8} \frac{10!}{5! 5!} < 2^{8} 252 < 256$

Therefore n=5 the given inequality is true.

Assume that the given inequality is true for n=k,

$(\begin{matrix} 2 k \\ k \end{matrix}) < 2^{2 k - 2}$

$\frac{(2 k)!}{(2 k - k)! k!} < 2^{2 k - 2}$

$\frac{(2 k)!}{k! k!} < 2^{2 k - 2}$ ...(1)

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