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Sequences (Induction?) Problem

  1. Jan 22, 2009 #1
    The problem statement, all variables and given/known data
    Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.

    a. Show that {an} is increasing and bounded.

    b. Prove that the limit as n approaches infinity of an exists.

    c. Find the limit as n approaches infinity of an.

    The attempt at a solution

    b is straightforward. If you show that an is monotonic and bounded then it has a limit.

    I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with an <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.
     
  2. jcsd
  3. Jan 23, 2009 #2
    a) calculate a(n+1)-a(n) and draw your conclusion.

    b)-

    c) In the limit for n tending to infinity you'll get: [tex]L = \sqrt{k+L} [/tex] which you can solve.
     
    Last edited: Jan 23, 2009
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