1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.


    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook