   Chapter 11.2, Problem 47ES ### 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

# a. Let x be any positive real number. Use mathematical induction to prove that for every integer m ≥ 1 , if x ≤ 1 then x m ≤ 1 . b. Explain how it follows from part (a) that if x is any positive real number, then for every integer m ≥ 1 , if then x > 1 . c. Explain how it follows from part (b) that if x is any positive real number, then for every integer m ≥ 1 , if x > 1 then x 1 / 3 > 1 .d. Let p, q, r, and s be positive integers, and suppose p / q > r / s . Use part (c) and the result of exercise 40 to prove Theorem 11.2.2. In other words, show that for any integer n, if n > 1 then n p / q > n r / s .

To determine

To prove:

k=3n(k22k) is Θ(n3)

Explanation

Given information:

k=3n(k22k)

Concept used:

Theorem of polynomial order.

Calculation:

k=3n( k 2 2k)=k=3nk22k=3nk=32+42+52+.....+n22[3+4+5+....+n]=(12+22+33+.....n21222)2(1+2+3+......+n12)=(12+22+32+......+n2)52(1+2+3+.....+n)+6=(12+22+32+......+n2)2(1+2+3+....

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