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Homework Help: Help proving with the Binomial Theorem

  1. Apr 30, 2010 #1
    1. The problem statement, all variables and given/known data

    (n¦0)-(n¦1)+(n¦2)-. . . ± (n¦n)=0

    that reads n choose zero and so on
    2. Relevant equations

    Prove this using the binomial theorem

    3. The attempt at a solution

    I really have no idea where to start. Any help would be greatly appreciated

    thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 30, 2010 #2
    Start by writing down the binomial theorem and seeing how you might get the expression you've typed in out of one side or the other.
     
  4. Apr 30, 2010 #3

    vela

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    What does the binomial theorem tell you? Start there.
     
  5. Apr 30, 2010 #4
    That as well.
     
  6. Apr 30, 2010 #5
    Well I've been staring at this thing for the past hour and I'm not coming up with anything. Am I to be looking at the (x+y)^n side of the binomial theorem or the side with the summation
     
  7. Apr 30, 2010 #6

    vela

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    Well, which side looks more like

    [tex]\begin{pmatrix}n\\0\end{pmatrix}-\begin{pmatrix}n\\1\end{pmatrix}+\cdots\mp\begin{pmatrix}n\\n-1\end{pmatrix}\pm\begin{pmatrix}n\\n\end{pmatrix}[/tex]
     
  8. Apr 30, 2010 #7
    steveT, if you write vela's sum expression in sigma notation, the result should jump right out at you.
     
  9. Apr 30, 2010 #8

    vela

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    You might also want to post what expression you have for the binomial theorem. There are different ways to write it, some more suggestive than others.
     
  10. Apr 30, 2010 #9
    This is the expression I'm using

    (x+a)^n=∑_(k=0)^n▒〖(n¦k) x^k a^(n-k) 〗
     
  11. Apr 30, 2010 #10

    vela

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    OK, so what values could you plug in for x and a such that you'd get the alternating sign but otherwise have them disappear?

    (It might help you to expand the summation to make the comparison more straightforward.)
     
  12. Apr 30, 2010 #11
    x=1 and a=0 ?
     
  13. Apr 30, 2010 #12
    Try plugging that into your equation for the binomial theorem and see what you get. What accounts for the alternating sign?
     
  14. Apr 30, 2010 #13
    I know the sigma notation can be a bit sketchy when one first learns it, so let me post this: What does this sum below equal? (according to the Binomial Theorem)

    [tex]\sum_{k=0}^n \binom{n}{k}(-1)^k[/tex]

    When dealing with sums the dot-dot-dots leave things a bit ambiguous. When you convert a sum with ... into something explicit using sigma notation, it usually makes things a lot easier.
     
  15. May 1, 2010 #14
    So when k is even you get plus and when its odd you get minus which accounts for the alternating sign.
     
  16. May 1, 2010 #15
    Expand the summation that Gauss^2 posted. What is it equal to in terms of (x+y)n? You are making this much harder than it has to be.

    You know that one side of the equation is zero. When is (x+y)n = 0? Use this along with the binomial theorem. Once you have your x and y, plug them into the formula for the binomial theorem to see if you do in fact get your desired alternating sum.
     
  17. May 1, 2010 #16
    Yes.
     
  18. May 3, 2010 #17
    Thanks everyone for your help. I UNDERSTAND
     
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