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Re-arranging equation.

  1. Oct 28, 2011 #1
    I need to rearrange the equation of km(m-1)+2m2 so it will look like km(m+1) in the end.

    I start by expanding the brackets:

    km(m-1)+2m2
    = km2-km+2m2
    = km(m-1)+2m2 (1)
    or m(km-k+2m) (2)
    or m2(2+k)-km (3)

    Then I'm stuck. I've tried many other ways but they all become either (1) or (2) or (3) at the end. I have to find a way to get the final result as km(m+1), please help!
     
  2. jcsd
  3. Oct 28, 2011 #2

    gb7nash

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    I hate to break it to you, but these two expressions are not algebraically equivalent.

    Let k = 0, m = 1.
     
  4. Oct 28, 2011 #3
    This is true, but if you mean to say you want the result to be km(m+1) + other stuff, then I suggest you try adding and subtracting km to the expanded expression, and then gathering km(m+1) from there.
     
  5. Oct 28, 2011 #4
    I must have done something wrong then. Will there be any solutions if k=(-1)m-1? I set k=(-1)m-1 to make life easier, maybe this is where I went wrong.

    So is there a way to arrange (-1)m-1m(m-1)+2m2 into (-1)m-1m(m+1)?
     
  6. Oct 28, 2011 #5

    gb7nash

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    Unfortunately, no. Try m = 3 for both and you'll obtain different answers for each.
     
  7. Oct 28, 2011 #6
    Can't figure where I've done wrong, so I thought I'd post the whole question and start by scratch.

    Q: Show the sum of the first m terms of the series 12 - 22 + 32 - 42 + ... is 0.5(-1)m-1m(m+1) (for m any positive integer).

    This is what I've done so far.

    12 - 22 + 32 - 42 + ... + (m-1)2
    = 0.5(-1)m-1-1(m-1)(m-1+1)
    = 0.5(-1)m-2(m)(m-1)

    Add -m2 to both sides:

    12 - 22 + 32 - 42 + ... + (m-1)2 - m2 = 0.5(-1)m-2(m)(m-1)-m2

    As the above equation represents the sum of the first m terms, so it must mean that
    0.5(-1)m-2(m)(m-1)-m2 = 0.5(-1)m-1m(m+1) and if I can prove this then I'm done.

    However, I figured that there's something wrong with the equation 0.5(-1)m-2(m)(m-1)-m2 as it's only true for when m is even. For any m that's odd, it doesn't work.

    So what exactly have I done wrong?
     
    Last edited: Oct 28, 2011
  8. Oct 28, 2011 #7
    Huh?
     
  9. Oct 28, 2011 #8
    Is it not correct? I sincerely don't know where I've gone wrong. If you have spotted any errors could you please let me know?
     
  10. Oct 29, 2011 #9

    eumyang

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    Here you're making the assumption that m is even. If m was odd, you would add +m2. I haven't tried to figure this out, but maybe you could add (-1)m-1m2 instead to take care of the signs.
     
  11. Oct 29, 2011 #10

    eumyang

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    I think I figured it out.
    You'll need the part that's bolded above as well. You're assuming that (m-1) is odd.
    As I've said, change to "add (-1)m-1m2 to both sides".
    12 - 22 + 32 - 42 + ... + (-1)m-2(m-1)2 + (-1)m-1m2 = 0.5(-1)m-2(m-1)(m) + (-1)m-1m2

    Now simplify the RHS to get 0.5(-1)m-1m(m+1).
     
    Last edited: Oct 29, 2011
  12. Oct 29, 2011 #11

    Mark44

    Staff: Mentor

    No, what you have above is the sum of the first m - 1 terms of the series. You are apparently doing a proof by induction, but you have not made any mention of this fact.
     
  13. Oct 29, 2011 #12

    Mark44

    Staff: Mentor

    eumyang, you have that backwards. The even terms are negative, and the odd terms are positive.
     
  14. Oct 29, 2011 #13
    I have managed to solve the problem using proof by induction, thanks for your help :)
     
  15. Oct 30, 2011 #14

    eumyang

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    I thought that was what I said previously. :confused:
     
  16. Oct 30, 2011 #15

    Mark44

    Staff: Mentor

    I think it actually was. I misinterpreted what you meant. I thought that you were saying referred to 12 - 22 + 32 -+ ... + m2, where m is even.
     
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