   Chapter 11.2, Problem 43ES ### 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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# Use Theorems 11.2.5-11.2.9 and the results of exercises 15-17, 40, and 41 to justify the statements in 43-45.43. ⌊ n + 1 2 ⌋ + 3 n is Θ ( n )

To determine

To show:

That the polynomial n+12+3n is θ(n).

Explanation

Given:

The polynomial n+12+3n.

Formula used:

The order of a polynomial is given by,

If m is any integer with m0 and a1,a2,...,am are real numbers with am>0, then amnm+am1nm1+...+a1n+a0 is θ(nm)

Proof:

Let p(n) be n+12+3n.

For all n0 we can define p(n) ,

p(n)={ n+1 2 +3n when n is even n+1 2 +3n when n is odd

When n is even, we can obtain that n2 is a complete number. Hence, n+12=n2

When

n is odd, n+1 is even

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