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Sequence/Induction question

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data

    A sequence xn is defined recursively by x1 = 2 and xn+1 = 8xn-1/4xn+3 for n ∈ N.
    a. Prove by induction or otherwise that xn -1 > 0 for all n ∈ N.

    2. Relevant equations


    3. The attempt at a solution

    x1 is given as 2, therefore x1 - 1 = 1 > 0, which satisfies the criteria.

    I'm not exactly sure how to employ induction (or the 'otherwise' method), from what I understand I need to prove that xn+1 - 1 > 0, as then the base case (n=1) and all other cases are proved... on the right track?
  2. jcsd
  3. Apr 14, 2010 #2


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    Is that middle term -(1/4)xn or -1/(4xn)?
  4. Apr 15, 2010 #3
    Yes, you seem to be on the right track.

    To use induction, you need to prove two things:

    1) that your statement is true for a particular case.
    2) that your statement being true for one case implies that it is true for the next case.

    I assume you meant to write (8xn - 1)/(4xn + 3) there.

    So 1) is easy, you already proved it.
    For 2), can you show that if xn > 1, xn+1 > 1 for all n [tex]\in[/tex] N?
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