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Natural Numbers and Induction (Analysis with and Introduction to ProofC)

  1. Jan 22, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove that: 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all n where n is a natural number

    2. Relevant equations

    Proof by induction:
    a) p(1) is true
    b) assume p(k) is true then prove p(k+1) for it

    3. The attempt at a solution

    I gave this a try for a while but I'm quite stuck.

    So far using proof by induction:
    p(1) = 1^3 = 1^2 is true

    assume that p(k) = 1^3 + 2^3 + ... + k^3 = (1 + 2 + ... + k)^2 is true
    then:
    p(k+1) = 1^3 + 2^3 + ... + k^3 + (k + 1)^3
    = (1 + 2 + ... + k)^2 + (k + 1)^3

    Now I'm really not sure where to go or if I'm going the right way. Any sort of hint would be greatly appreciated. Thanks.

    -Kentt
     
  2. jcsd
  3. Jan 22, 2007 #2
    Simplify 1+2+3+...k.
     
  4. Jan 23, 2007 #3

    Gib Z

    User Avatar
    Homework Helper

    [tex]\sum_{n=1}^k n= \frac{(k+1)(k)}{2}[/tex] btw.
    You may also like to know that [tex]\sum_{n=1}^k n^3 = \frac{(k+1)^2k^2}{4}=(\frac{(k+1)k}{2})^2[/tex]...o nvm Thats what your trying to prove...
     
  5. Jan 23, 2007 #4
    Thanks IMDerek. Got it done.
     
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