Help proving with the Binomial Theorem

[steveT]

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

 

[Martin Rattigan]

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.

 

[vela]

What does the binomial theorem tell you? Start there.

 

[Martin Rattigan]

That as well.

 

[steveT]

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

 

[vela]

Well, which side looks more like

$$\begin{pmatrix}n\\0\end{pmatrix}-\begin{pmatrix}n\\1\end{pmatrix}+\cdots\mp\begin{pmatrix}n\\n-1\end{pmatrix}\pm\begin{pmatrix}n\\n\end{pmatrix}$$

 

[gauss^2]

steveT, if you write vela's sum expression in sigma notation, the result should jump right out at you.

 

[vela]

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.

 

[steveT]

This is the expression I'm using

(x+a)^n=∑_(k=0)^n▒〖(n¦k) x^k a^(n-k) 〗

 

[vela]

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.)

 

[steveT]

x=1 and a=0 ?

 

[VeeEight]

Try plugging that into your equation for the binomial theorem and see what you get. What accounts for the alternating sign?

 

[gauss^2]

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)

$$\sum_{k=0}^n \binom{n}{k}(-1)^k$$

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.

 

[steveT]

So when k is even you get plus and when its odd you get minus which accounts for the alternating sign.

 

[VeeEight]

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.

 

[gauss^2]

So when k is even you get plus and when its odd you get minus which accounts for the alternating sign.

Yes.

 

[steveT]

Thanks everyone for your help. I UNDERSTAND