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.

44. n ( n − 1 ) 2 + ⌊ n 2 ⌋ + 1 is Θ ( n 2 )

To determine

To show:

That the polynomial n(n−1)2+⌊n2⌋+1 is θ(n2).

Explanation

Given:

The polynomial n(n−1)2+⌊n2⌋+1.

Formula used:

The order of a polynomial is given by,

If m is any integer with m≥0 and a1,a2,...,am are real numbers with am>0, then amnm+am−1nm−1+...+a1n+a0 is θ(nm)

Proof:

Let p(n) be n(n−1)2+⌊n2⌋+1.

For all n≥0 we can define p(n) ,

p(n)={ n( n−1 ) 2+⌊ n 2 ⌋+1 when n is even n( n−1 ) 2+⌊ n 2 ⌋+1 when n is odd

For n is even, we can obtain that n2 is an integer. Hence, ⌊n2⌋=n2

Also, when n is odd, n2 is not an integer. But, n−12 is an integer such that n2≥n−12

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