# The Binomial Theorem

1. May 26, 2005

### Cincinnatus

It seems like this shouldn't be too difficult and yet i'm stumped.
I am trying to prove the binomial theorem.

(x+y)^n = the sum from k=0 to n of (x^k)*(y^n-k)*(The binomial coefficient n,k)

Anyway, I figure the best way to go about proving this is by induction.
It is easy to show that its true for n=1.
Then I assume that there exists an n_0 such that it is true for all n < n_0.
Now I want to show that the existence of this n_0 implies that the proposition is also true for n=n_0+1.

This is where I get stuck...
My question is, is this even the best way to go about proving this? If so, how can I finish the proof?

Maybe it would be better to give me a hint so I can figure it out on my own...

2. May 26, 2005

### Dr.Brain

3. May 27, 2005

### AKG

What you need to do is look at Pascal's Triangle. You should be aware of the simple procedure required for you to recreate the triangle yourself. You have to show that the number in one spot is equal to the sum of the two numbers above it. Now assume as your inductive hypothesis that the binomial expansion works for the exponent k. To prove that it works for exponent k+1, multiply the assumed expansion for k by just (a - b) (or whatever you are using for the base where the exponent is k), and use the facts just mentioned about Pascal's triangle to show that the expansion takes the desired form for k+1 as well.

EDIT: which is precisely what PlanetMath seems to tell you.

Last edited: May 27, 2005