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Homework Help: Find a recurrence formula

  1. Nov 30, 2017 #1
    1. The problem statement, all variables and given/known data
    Find a recurrence formula for the sequence (ai) = 1, sqrt3, sqrt(1+sqrt3), sqrt(1+sqrt(1+sqrt2)) in terms of i and ai

    2. Relevant equations


    3. The attempt at a solution
    no idea where to start, this is a bonus question, and I have learnt how to solve these type of problems
     
  2. jcsd
  3. Nov 30, 2017 #2

    jedishrfu

    Staff: Mentor

    I'd investigate nested radicals as this was a favorite of Ramanujan.

    As a hint,## x = \sqrt{1 + \sqrt{1+ \sqrt{1 + ...}}} ## is basically the same as ## x = \sqrt{1 + x}##
     
  4. Nov 30, 2017 #3
    That hasn't helped. Please explain.
     
  5. Nov 30, 2017 #4

    Mark44

    Staff: Mentor

    Shouldn't the last one you listed be ##\sqrt{1 + \sqrt{1 + \sqrt 3}}##?
    Start by listing the elements of your sequence in an organized fashion, like so:
    ##a_0 = 1##
    ##a_1 = \sqrt 3##
    ##a_2 = \sqrt{1 + \sqrt 3}## What is ##a_2## in terms of ##a_1##?
    ##a_3 = \sqrt{1 + \sqrt{1 + \sqrt 3}}## What is ##a_3## in terms of ##a_2##?
    Can you predict what ##a_4## is? If you can, you might be able to write ##a_n## in terms of ##a_{n - 1}##, which is what you need to do for this problem.
     
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