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Introductory Analysis: Inductively define a sequence Sn

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data


    Let S1=1 and inductively define the sequence Sn so that Sn+1 = [tex]\sqrt{Sn + 1}[/tex]


    2. Relevant equations



    3. The attempt at a solution

    I'm not sure what it means to "inductively define".

    I think it wants me to come up with an equation for Sn by using Sn+1.

    Does it want me to define Sn in terms of Sn+1 or just in terms of n?

    How should I go about starting this?
     
  2. jcsd
  3. Jan 25, 2009 #2
    Surely this isn't the complete assignment . Please post the entire question only then I can help you.
     
  4. Jan 25, 2009 #3
    Let S1=1 and inductively define the sequence (Sn) so that Sn+1 = [tex]\sqrt{Sn + 1}[/tex] for n[tex]\in[/tex] Natural Numbers.

    (a) Prove that Sn is a monotonically increasing sequence.
    (b) Prove that Sn is a bounded sequence.
    (c) Prove that Sn converges.
    (d) Prove that lim(Sn)=[tex]\frac{1}{2}[/tex] (1 + [tex]\sqrt{5}[/tex] )
     
  5. Jan 25, 2009 #4
    I'm sorry. That's the whole thing now.

    I didn't realize that you needed the a,b,c,d parts to do the first part, I thought you had to inductively define Sn and then, using that definition, do the rest.
     
  6. Jan 25, 2009 #5

    Dick

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    'Inductively define' doesn't mean you have to do anything. It's just pointing out that S_{n+1}=sqrt(S_n+1) is already an 'inductive' definition.
     
  7. Jan 25, 2009 #6
    Oh, haha. Thank you.
     
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