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Homework Help: Mathematical Induction problem

  1. Aug 22, 2010 #1
    I'm trying to solve this problem from CH2 of spivak's calculus of which I am self-studying.

    1. The problem statement, all variables and given/known data
    Prove the following by mathematical induction:
    [tex]1^3+...+n^3=(1+...+n)^2[/tex]


    2. Relevant equations
    To prove by mathematical induction, you test whether P(1) is true and if P(k) is true then P(k+1) is true.


    3. The attempt at a solution
    [tex]1^3+...+n^3=(1+...+n)^2[/tex]
    [tex]1^3+...+n^3+(n+1)^3=(1+...+n)^2+(n+1)^3[/tex]
    [tex](1+...+n)^2+(n+1)^3=(1+...+n+1)^2[/tex]

    From there I have no clue and I've been staring at that for 15 minutes.
     
  2. jcsd
  3. Aug 22, 2010 #2

    Mentallic

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    Homework Helper

    Don't forget that the RHS is

    [tex](1+2+...+n+(n+1))^2[/tex]

    (I'm just making sure that we are on the same page here, since what you wrote can be mistaken as missing that extra n).

    Now take the RHS and treat 1+2+...+n as one constant on its own, and expand.
     
  4. Aug 22, 2010 #3

    hunt_mat

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    Homework Helper

    Show that it's true for n=1 (or n=2 even)
    1^3=1^2, true for n=1, 1^3+2^3=9=(1+2)^2=3^2=2, so true for n=2. Suppose there is a k such that:
    [tex]
    1^{3}+2^{3}+\cdots +k^{3}=(1+2+\cots +k)^{2}
    [/tex]
    So, we write:
    [tex]
    (1+2+\cdots +k+(k+1))^{2}=(1+2+\cdots +k)^{2}+(k+1)^{2}+2(k+1)(1+2+\cdots +k)
    [/tex]
    Now you should know a formula for (1+2+...+k), the equation becomes:
    [tex]
    (1+2+\cdots +k+(k+1))^{2}=1^{3}+2^{3}+\cdots +k^{3}+(k+1)^{2}+2(k+1)(1+2+\cots +k)
    [/tex]
    So you now have to show that:
    [tex]
    (k+1)^{3}=(k+1)^{2}+2(k+1)(1+2+\cdots +k)
    [/tex]
    How would you go about doing this?
     
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