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Inductive Proof

  1. Mar 9, 2004 #1
    I'm stuck and not sure what I've done wrong for this problem

    Prove the following by mathematical induction:
    1^3 + 2^3 + 3^3 + 4^3 ..... + n^3 = ((n^2 + n)/2)^3

    ok so I proved it for n = 1 and n = 2 then assume n = k
    so ((k^2 + k)/2)^3

    Then lets try to do n = k +1
    so
    ((k+1)^2 + k +1)/2)^3 = ((k^2 + k)/2)^3 + (k + 1)^3
    after expanding I get

    (k^6+ 9k^5+ 33k^4+ 63k^3+ 66k^2+ 36k+ 8)/8
    =
    (k^6 +3k^5 + 3k^4 + 9k^3 + 24k^2 + 24k + 8)/8 + k^3 + 3k^2 + 3k +1

    For some reason I think that I don't have to do all of this expanding. Can someone please tell me what I"m doing wrong or what I need to fix?
     
  2. jcsd
  3. Mar 9, 2004 #2
    Nevermind I just heard from a fellow friend that this statement is false.
     
  4. Mar 12, 2004 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Someone had to tell you that? Did you even try calculating a few numbers?

    If n= 2, 12+ 22= 1+ 8= 9

    [tex]\(\frac{n^2+n}{2}\)^3= \(\frac{4+2}{2}\)^3= 3^3= 27[/tex]

    so for the very second number its not true.

    If you had done even the slightest amount of work on this you would have seen:
    1 13 = 1
    2 13+23= 1+ 8= 9
    3 13+23+33= 1+ 8+ 27= 36
    4 13+23+33+43= 100.

    Hmmm: 1, 9, 36, 100,... what does that make you think of? Not cubes certainly!
     
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