   Chapter 11.4, Problem 21ES ### 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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# In each of 21 and 22, a sequence is specified by a recurr ence relation and initial conditions. In each case, (a) use iteration to guess an explicit formula for the sequence; (b) use strong mathematical induction to confirm the corr ectness of the formula you obtained in part (a).21. a k = a ⌊ k / 2 ⌋ + 2 , for each integer k ≥ 2 a 1 = 1

To determine

(a)

Use iteration to guess an explicit formula for the sequence, ak=ak/2+2 for all integers k2,

a1=1 ..

Explanation

Given information:

A sequence is specified by a recurrence relation and initial conditions.

ak=ak/2+2, for all integers k2, a1=1

Calculation:

We have ak=ak/2+2, for all integers k2, a1=1

Let us determine the first terms of the given recurrence relation:

a1=1=1+2log21

a2=a2/2+2=a1+2=1+2=3=1+2log22

a3=a3/2+2=a1+2=1+2=3=1+2log23

a4=a4/2+2=a2+2=3+2=5=1+2log24

a5=a5/2+2=a2+2=3+2=5=1+2log25

a6=aȰ

To determine

(b)

Use strong mathematical induction to confirm the correctness of the formula we obtain in part (a).

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