   Chapter 11.2, Problem 39ES ### 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—39. Use the theorem on polynomial orders and results from the theorems and exercises in Section 5.2 as appropriate.39. ∑ k = 3 n ( k 2 − 2 k ) is Θ ( n 3 )

To determine

To prove the following statement using the theorem on polynomial orders and results.

Explanation

Given information:

k=3n(k22k)is Θ(n3)

Calculation:

Expanding the direct summation of the series-

k=3 n ( k 2 2k)=( 3 2 23+ 4 2 24+ 5 2 25+....... n 2 2n)

=( 3 2 + 4 2 + 5 2 +.......+ n 2 )(23+24+25....+2n)

{Adding and subtracting 1 2 +2 2  in first bracket and 21+22 in second bracket}

=[( 1 2 + 2 2 + 3 2 + 4 2 + 5 2 +.......+ n 2 )( 1 2 + 2 2 )][(21+22+23+24+25....+2n)(21+22)]

=[( 1 2 + 2 2 + 3 2 + 4 2 + 5 2 +.......+ n 2 )( 1 2 + 2 2 )]2[(1+2+3+4+5

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