   Chapter 11.3, Problem 27ES ### 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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# Consider the recurrence relation that arose in Example 11.3.7: E 1 = 0 and E k − 1 + k + 1 2 , for each integer k ≥ 2 . a. Use iteration to find an explicit formula for the sequence. b. Use mathematical induction to verify the correctness of the formula.

To determine

(a)

Use iteration to find an explicit formula for the sequence.

Explanation

Given information:

Consider the recurrence relation that arose in example 11.3.7: E1=0and Ek=Ek1+k+1 for all integers k2.

Concept used:

Apply iteration method to find an explicit formula for the sequence.

Calculation:

By the method of finding an average-case order for insertion sort and by the given data,

E1=0and Ek=Ek1+k+1, for all integers k2.

Now,

E2=E1+2+1=0+2+1since E1=0=3by algebra

E3=E2+3+1=3+3+1since E2=3=3+4

E4=E3+4+1=3+4+4+1sinceE3=3+4=3+4+5

En=3+4+5+..........+(n1)since by iteration

=[1+2+3+4+

To determine

(b)

Use mathematical induction to verify the correctness of the formula.

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