   Chapter 11.4, Problem 51ES ### 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
1 views

# Complete the proof in Example 11.4.4.

To determine

Complete the proof in Example 11.4.4.

Explanation

Given information:

ak=2ak/2 for all integers k2a1=1

Proof:

To prove: an=2log2n for all integers n1

PROOF BY STRONG INDUCTION:

Let P(n) be "an=2log2n"

Basis step: n = 1

a1=1=20=2log21

Thus P (1) is true.

INDUCTIVE STEP:

Let P(1),P(2),...,P(k) be true, thus ai=2log2i for i=1,2,...,k

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.

ak+1=2a(k+1)/2                                                     ak=2ak/2

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

=22log2(k+1)/2                                   P((k+1)/2) is true

=2log2(k+1)/2+1

=2log2(k+1)log22+1                             logbxy=logbxlogby

=2log2(k+1)1+1

=2log2(k+1)1+1

=2log2(k+1)

SECOND CASE: k even

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

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