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Mathematical Induction using a strong hypothesis

  1. Dec 4, 2008 #1
    1. The problem statement, all variables and given/known data
    If a0=1, and a1=2, and

    an=(a(n-1))^2/an-2 for n>=2,
    prove by induction that an=2^n for n>=0



    2. Relevant equations



    3. The attempt at a solution
    (B) a0=1=2^0=1 yes is true
    a1=2=2^1=2 yes is true

    (I) ak=(2k-1)^2/2k-2=2k

    Is it true that what I solved. It seems very easy.

    I started first with a(k+1)=(a(k+1-1))^2/a(k+1-2)=
    (2k)^2(k-1)=2(k+1)
    Please someone help if I am doing something wrong. I will appreciate. Thank you.
     
  2. jcsd
  3. Dec 4, 2008 #2
    It seems that you have the right idea, but I do not know what is different between (I) and your solution for ak+1. Where precisely is your inductive hypothesis?
     
  4. Dec 4, 2008 #3

    Dick

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    Science Advisor
    Homework Helper

    It IS very easy. So your induction step is: if a_{k-1}=2^(k-1) and a_{k-2}=2^(k-2) then a_k=2^(k) (by doing the algebra you doubtless did). It looks fine to me.
     
  5. Dec 4, 2008 #4
    That is where I am confused. I know that I need to fallow the form an=2^n

    Then I don't need to do a(k+1)....

    I did so many problems in my HW and now I don't get it what I am doing:smile:
     
  6. Dec 4, 2008 #5
    Thank you guys.
     
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