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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:

    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

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


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    Don't forget that the RHS is


    (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


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    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:
    1^{3}+2^{3}+\cdots +k^{3}=(1+2+\cots +k)^{2}
    So, we write:
    (1+2+\cdots +k+(k+1))^{2}=(1+2+\cdots +k)^{2}+(k+1)^{2}+2(k+1)(1+2+\cdots +k)
    Now you should know a formula for (1+2+...+k), the equation becomes:
    (1+2+\cdots +k+(k+1))^{2}=1^{3}+2^{3}+\cdots +k^{3}+(k+1)^{2}+2(k+1)(1+2+\cots +k)
    So you now have to show that:
    (k+1)^{3}=(k+1)^{2}+2(k+1)(1+2+\cdots +k)
    How would you go about doing this?
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