Help proving with the Binomial Theorem

  • Thread starter steveT
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  • #1
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Homework Statement



(n¦0)-(n¦1)+(n¦2)-. . . ± (n¦n)=0

that reads n choose zero and so on

Homework Equations



Prove this using the binomial theorem

The Attempt at a Solution



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

thanks

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #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.
 
  • #3
vela
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What does the binomial theorem tell you? Start there.
 
  • #5
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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
 
  • #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]
 
  • #7
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steveT, if you write vela's sum expression in sigma notation, the result should jump right out at you.
 
  • #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.
 
  • #9
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This is the expression I'm using

(x+a)^n=∑_(k=0)^n▒〖(n¦k) x^k a^(n-k) 〗
 
  • #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.)
 
  • #11
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x=1 and a=0 ?
 
  • #12
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Try plugging that into your equation for the binomial theorem and see what you get. What accounts for the alternating sign?
 
  • #13
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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.
 
  • #14
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So when k is even you get plus and when its odd you get minus which accounts for the alternating sign.
 
  • #15
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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.
 
  • #16
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So when k is even you get plus and when its odd you get minus which accounts for the alternating sign.

Yes.
 
  • #17
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Thanks everyone for your help. I UNDERSTAND
 

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