Answered: Use induction to show that: floor(n/2)…

Name: Use induction to show that: floor(n/2) + ceil(n/2) = n
Uploaded: 2024-03-20T06:46:55.000Z
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Description: Use induction to show that: floor(n/2) + ceil(n/2) = n

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use induction to show that: floor(n/2) + ceil(n/2) = n

Expert Solution

Step 1

Check for n= 1

floor(1/2) + ceil(1/2) = 0 +1 =1

So it is true for n= 1

Let us assume that it is also true for n =k

floor(k/2) + ceil(k/2) = k ... ...(1)

And try to show if it is true for n= k+1 too.

So for n= k+1

floor((k+1)/2) + ceil((k+1)/2)

floor((k/2)+floor(1/2) + ceil((k/2)+ceil(1/2)

floor((k/2)+ ceil((k/2)+0+1

k+1 using statement (1)

Hence it is true for n= k+1 also so it is true for all natural numbers.

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