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Homework Help: Prove by Induction

  1. Mar 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove by induction

    1^2 + 2^2 + ... + (n-1)^2 < (n^3)/3

    (This is the problem on page 33 of Apostol's book)



    2. Relevant equations



    3. The attempt at a solution

    A(k) = 1^2 + 2^2 + ... + (k-1)^2 < (k^3)/3

    A(k+1) = 1^2 + 2^2 + ... + k^2 < (k+1)^3/3


    Start with A(k) and add k^2 to both sides. This gives the inequality

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2


    To obtain A(k+1) as a consequence of this, it suffices to show that

    k^3/3 + k^2 < (k+1)^3 /3




    Now (k+1)^3 /3 = k^3/3 + k^2 +k +1/3 which is greater than k^3/3 + k^2


    Which proves the inequality. However I dont understand why does

    k^3/3 + k^2 < (k+1)^3 /3 suffice to prove the inequality

    Can anyone explain in detail why does that inequality proves it?
     
  2. jcsd
  3. Mar 13, 2010 #2

    vela

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    They're using the transitive property of <. You know

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2.

    If you can show (k^3)/3 + k^2 < (k+1)^3/3, then you'll have

    1^2 + 2^2 + ... + k^2 < (k^3)/3 + k^2 < (k+1)^3/3

     
  4. Mar 13, 2010 #3
    Thanks a lot, I really appreciate it. :) I have been struggling with this for quite some time.
     
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