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REAL ANALYSIS, Math Induction

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    The problem and my solution attempt are in the attached file.
    Am I doing it right? I didn't write the final answer because it is not what I expected. Just wanted to hear if I made any mistakes. Thank you.


    2. Relevant equations



    3. The attempt at a solution
     

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  2. jcsd
  3. Aug 29, 2010 #2
    Here is what I get...I am unable to bring it to an original formula...
    Is it right that we cannot prove the expression?
     

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  4. Aug 29, 2010 #3
    One sec, I see my mistake.
     
  5. Aug 29, 2010 #4

    hunt_mat

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    You should be adding (2(k+1)-1)^2=(2k+1)^2 to each side of the equality
     
  6. Aug 29, 2010 #5
    OK, here is my corrected version.
    My final answer is bulky. I tried to open brackets but all I get is
    (4k^3+4k^2+3k+1)/3.

    Please help me from here. Or did I make a mistake before?
     

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  7. Aug 29, 2010 #6

    hunt_mat

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    You forgot to multiply by 3 when you put everything over 3.
     
  8. Aug 29, 2010 #7
    I don't understand, sorry.
     
  9. Aug 29, 2010 #8

    hunt_mat

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    [tex]
    \frac{4k^{3}-k}{3}+(2k+1)^{2}=\frac{4k^{3}-k+3(2k+1)^{2}}{3}
    [/tex]
     
  10. Aug 29, 2010 #9
    Thank you for the correction.
    Where do I go from there? I tried open the brackets. Got
    4k^3+12k^2+11k+3. Doesn't look nice.
     

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  11. Aug 29, 2010 #10

    hunt_mat

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    Take out a factor of 4 and ask yourself what the exapnsion of (k+1)^3 is.

    Mat
     
  12. Aug 29, 2010 #11
    If I open brackets, I get
    4k^3+12k^2+11k+3.
    I cannot factor it by 4.
     
  13. Aug 29, 2010 #12
    Stop at [tex] = \frac{4k^3 - k + 3(2k + 1)^2}{3} [/tex] and expand the numerator completely. You know you want 4(k+1)^3. So do as hunt_mat suggested and expand 4(k+1)^3 as an aside (not in the proof) so you know what it is expanded. Subtract this expanded form from your expanded numerator. It should work.
     
  14. Aug 29, 2010 #13
    Thank you!
    It turned out very well.
     

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