   Chapter 11.4, Problem 32ES ### 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.32. 1 + 2 + 2 2 + 2 3 + ⋯ + 2 n is Θ ( 2 n ) .

To determine

To prove:

1+2+22+23+...+2n is Θ(2n).

Explanation

Given information:

assume n is an integer variable that takes positive integer values.

Proof:

1+2+22+23+...+2n=k=0n2k

=k=0(n+1)12k

=2n+1121                                    k=0n1ak=an1a1

=2n+111

=2n+11

Since 2n+1=22n and 10:

|2n+11||2n+1|=2|2n| whenever n0

However, we also know that 2n2n+1

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