   Chapter 11.5, Problem 14ES ### 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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# Exercises 12—15 refer to the following algorithm segment. For each positive integer n, let b n be the number of iterations of the while loop. while   ( n > 0 ) n : = n   d i v   3 end while 14. a. Use iteration to guess an explicit formula for b n . b. Prove that if k is an integer and x is a real number with 3 k ≤ x < 3 k then ⌊ log 3 x ⌋ .c. Prove that for every integer m ≥ 1 ,   ⌊ log 3 ( 3 m ) ⌋ =   ⌊ log 3 ( 3 m + 1 ) ⌋ = ⌊ log 3 ( 3 m + 2 ) ⌋ .d. Prove the correctness of the formula you found in part (a).

To determine

(a)

Use iteration to guess an explicit formula for bn.

Explanation

Given information:

For each positive integer n, let bn be the number of iterations of the while loop. while ( n > 0)

n := n div 3

end whileCalculation:

Result previous exercise:

bn=bn/3+1 for all integers n2b1=1b2=1

Let us determine the first terms of the given recurrence relation:

b1=1=1+log31

b2=1=1+log32

b3=b3/3+1=b1+1=1+1=2=1+log33

b4=b4/3+1=b1+1=1+1=2=1+log34

b5=b5/3+1=b1+1=1+1=2=1+log35

b6=b6/3+1=b2+1=1+1=2=1+log3</

To determine

(b)

Prove that if k is an integer and x is a real number with 3kx<3k+1, then log3x=k.

To determine

(c)

Prove that for all integers m1,

log3(3m)=log3(3m+1)=log3(3m+2).

To determine

(d)

Prove the correctness of the formula you found in part (a).

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