   Chapter 11.4, Problem 23ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Define a sequence c 1 ,   c 2 ,   c 3 ,   … recursively as follows: c 1 = 0 c k = c ⌊ k / 2 ⌋ + k , for each integer k ≥ 2 .Use strong mathematical induction to show that c n ≥ n 2 for every integer n ≥ 1 .

To determine

To prove:

Show that cnn2 for all integers n1.

Explanation

Given information:

Define a sequence c1,c2,c3,...., recursively as follows:

c1=0ck=2ck/2+k, for all integers k2.

Proof:

PROOF BY STRONG INDUCTION:

Let P(n) be "cnn2"

Basis step: n = 1

cn=c1=01=12=n2

Thus P (1) is true.

INDUCTIVE STEP:

Let P(1),P(2),...,P(k) be true, thus ci=i2 for i=1,2,...,k and let k2.

We need to prove that P(k+1) is true.

First case: k odd

Since k is odd, k + 1 is even and thus (k+1)/2 is an integer.

ck+1=2c(k+1)/2+k                                  ck=2ck/2+k

=2c(k+1)/2+2                                (k+1)/2 is an integer

2(( k+1)/2)2+2                         P((k+1)/2) is true

=2( k+1)24+2

=k2+2k+12+2                             (a+b)2=a2+2ab+b2

=k22+k+32

k22+k22+32                                         2kk2 when k2

=k2+32

k2+2             &

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