Proving the Binomial Theorem with Induction

In summary, the conversation is about trying to prove the binomial theorem using induction. The method involves looking at Pascal's Triangle and showing that the expansion works for both k and k+1 by multiplying the assumed expansion for k by (a - b) and using the properties of the triangle. The link provided also explains this method.
  • #1
Cincinnatus
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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)

Sorry, about the notation...

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...
 
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  • #2
http://planetmath.org/encyclopedia/InductiveProofOfBinomialTheorem.html

Go to the above link
 
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  • #3
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. :redface:
 
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1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that shows the expansion of a binomial expression raised to a positive integer power. It is commonly written as (a + b)^n = a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + ... + nCn-1ab^(n-1) + b^n, where n is a positive integer and nCk represents the binomial coefficient.

2. How is the Binomial Theorem proven?

The Binomial Theorem can be proven using mathematical induction, which is a method of mathematical proof that involves showing that a statement holds true for a base case and then using that to prove that it holds true for the next case. In this case, we use induction to show that the formula holds true for all positive integers n.

3. What is mathematical induction?

Mathematical induction is a method of mathematical proof that is used to prove statements about integers. It involves proving that a statement holds true for a base case (usually n = 1) and then showing that if it holds true for n, it also holds true for n+1. This allows us to prove that the statement holds true for all positive integers.

4. Why is induction used to prove the Binomial Theorem?

Induction is used to prove the Binomial Theorem because it is a powerful tool for proving statements about integers. The Binomial Theorem involves expanding binomial expressions, which are made up of integer coefficients, and raising them to positive integer powers. Induction allows us to prove that the formula holds true for all possible values of n, making it the most efficient method for proving the Binomial Theorem.

5. What are the steps for proving the Binomial Theorem with induction?

The steps for proving the Binomial Theorem with induction are as follows:

  1. Base case: Show that the formula holds true for n = 1 (i.e. (a + b)^1 = a + b).
  2. Inductive hypothesis: Assume that the formula holds true for n = k (i.e. (a + b)^k = a^k + kC1a^(k-1)b + kC2a^(k-2)b^2 + ... + kCk-1ab^(k-1) + b^k).
  3. Inductive step: Show that if the formula holds true for n = k, it also holds true for n = k + 1. This involves expanding (a + b)^(k + 1) using the inductive hypothesis and showing that it is equivalent to the formula for n = k + 1.
  4. Conclusion: By the principle of mathematical induction, we can conclude that the formula holds true for all positive integers n.

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