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Proof by induction of a sequence.

  • Thread starter missavvy
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  • #1
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Homework Statement



Given a sequence of rational numbers, defined inductively as follows:

s1 = 1, sn+1 = sn/2 + 1/sn, n>=1

prove that 1<=sn<=2 forall n>=1

Homework Equations





The Attempt at a Solution


I've got the solution to this but I don't understand a certain part, I was hoping someone could explain it to me?

So Let P(n) be the statement 1<=sn<=2
P(1) is satisfied.
Suppose P(n) holds.
Then sn+1 = sn/2 + 1/sn >= 1/2 + 1/2 = 1, because sn/2>=1/2 and 1/sn<=1/2

The bold part is what i don't understand. Wouldn't that imply that sn >= 2?
Am I mistaken or is there maybe an error in this solution?

Thanks in advance.
 

Answers and Replies

  • #2
Office_Shredder
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The inequality is written backwards. It should read [tex] \frac{1}{s_n} \geq \frac{1}{2}[/tex], which of course is what you need to write the >= earlier in that line
 
  • #3
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thanks I thought so... But then this solution continues as follows.....

Moreover, Sn+1 = sn/2 + 1/sn <= 2/2 + 1/2 <= 3/2 <= 2

I'm unsure about this, because then wouldn't you need 1/sn <= 1/2 for this step?

____________________


Ok forgetting about that solution, if I were to say:

Sn+1 = Sn/2 + 1/Sn >= 1/2 + 1/2 = 1
and
Sn+1 = sn/2 + 1/Sn <= 1 + 1 = 2

Is this right and proves that P(n+1) holds?
 
Last edited:
  • #4
You can use AM-GM inequality.
 

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