   Chapter 11.4, Problem 34ES ### 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.34. 2 + 2 ⋅ 3 2 + 2 ⋅ 3 4 + ⋯ + 2 ⋅ 3 2 n is Θ ( 3 2 n ) .

To determine

To prove:

2+2.32+2.34+...+2.32n is  Θ(32n) by assuming n is a positive integer variable.

Explanation

Definition used:

f is of order g: f(x) is Θ(g(x)) if there exists a positive real numbers, A, B and a nonnegative real number k such that A|g(x)||f(x)|B|g(x)| whenever x>k

Proof:

2+2.32+2.34+...+2.32n =2(1+32+34+...+3 2n )=2(1+9+92+...+9n )=2k=0n9k=2k=0(n+1)19k=2.9 n+1191=9 n+114=94.9n14

Since 14<0, |94.9n14||94.9n|=94.|9n| whenever n0

Since, 9n9n+11 as n0, |94

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