   Chapter 11.4, Problem 36ES ### 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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# Prove each of the statements in 32—37, assuming n is an integer variable that takes positive integer values. Use identities from Section 5.2 as needed.36. n + n 2 + n 4 + ⋯ + n 2 n is Θ ( n ) .

To determine

To prove:

n+n2+n4+...+n2n is Θ(n).

Explanation

Given information:

assumen is an integer variable that takes positive integer values.

Proof:

n+n2+n4+...+n2n=n(1+12+122+...+12n)                       Distributive property

=n(1+12+( 1 2 )2+...+( 1 2 )n)

=nk=0n( 1 2 )k

=nk=0(n+1)1( 1 2 )k

=n( 1 2 )n+11121                                       k=0n1ak=an1a1

=n( 1 2 )n+1112

=n(2( 1 2 )n)

Since (12)n0:

|n(2( 1 2 )n)||n(2+0)|=|n2|=2|n| whenever n0

However, we also know that 12(12)n when n1 and thus 12(12)n when  n1 (since (12)n decreases as n increases)

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