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verifying the inequality using induction

Posted on 2006-06-11
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I have a question about verifying the inequality using induction.

2N + 1<or =  2^N,       N = 3, 4,....      

So far, I have the basis
2(3) + 1 < or = 2^3
6 + 1 < or = 8                                    
7 < or = 8

Trying to do the induction step and it is not making much sense. I am assuming that since I am using 3 in the basis, I am using n+3 in the inductive? The problem I think I am having the most trouble with is the power of.
This is what I have, although I am sure it is completely incorrect.

2(n+3) + 1 < or = 2^n+3
2n + 6 + 1 < or = 2^n+3
2n + 7 < or = 2^n+3

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Question by:rcanter
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Harisha M G earned 500 total points
ID: 16881566
Hi, you have proved it for N=3. Now, put N=N+1, and prove that

2(N+1) + 1 <=  2^(N+1)

Now, you have

2N + 1 <=  2^N

Multiply both sides by 2

4N + 2 <= 2^(N+1)

2(N+1) + 2N <= 2^(N+1)

Since N is positive, 2N is also positive. Also, N >= 3. Hence, 2N >= 6 and obviously 2N >= 1, so we can write,

2(N+1) + 1 <= 2(N+1) + 2N <= 2^(N+1)

Removing the intermediate term,

2(N+1) + 1 <= 2^(N+1)

Hence the proof


---
Harish
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Author Comment

by:rcanter
ID: 16881572
U are awesome!!! Thanks so much!!!
One quick question, just so that I understand.  Even that I used S(3) for the basis, I am using S(n+1) for the inductive?
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Expert Comment

by:Harisha M G
ID: 16881587
rcanter, yes. You used S(3) because it is given in the problem itself that N = 3,4...

As I told in the previous question, you need to prove it that the (in)equality holds for atleast one value, which in case if it is not given, will be taken generally as 1
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Author Comment

by:rcanter
ID: 16881614
I am not understanding how this could be proven.
2N + 1 <=  2^N

Multiply both sides by 2     ***why am i multiplying by 2?***

4N + 2 <= 2^(N+1)

2(N+1) + 2N <= 2^(N+1)     **why wouldnt this be 2(n+1) + 1? how did 2n figure in?**

Since N is positive, 2N is also positive. Also, N >= 3. Hence, 2N >= 6 and obviously 2N >= 1, so we can write,

2(N+1) + 1 <= 2(N+1) + 2N <= 2^(N+1)

Removing the intermediate term,

2(N+1) + 1 <= 2^(N+1)

Hence the proof
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LVL 37

Expert Comment

by:Harisha M G
ID: 16881703
you need to get N+1 in either side and then prove the other side
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