Proving the Evenness of (\stackrel{2n}{n}) Using the Binomial Theorem

In summary, the conversation discusses how to prove that (\stackrel{2n}{n}) is even for n \geq 1 using the given identity and inductive reasoning. The solution involves showing that (\stackrel{2n}{n}) is equal to (2) (\stackrel{2n-1}{n-1}), which is always an integer.
  • #1
ipitydatfu
14
0

Homework Statement



prove that ([tex]\stackrel{2n}{n}[/tex]) is even when n [tex]\geq1[/tex]

Homework Equations



as a hint they gave me this identity:
[tex]\stackrel{n}{k}[/tex]= (n/k)([tex]\stackrel{n-1}{k-1}[/tex])

The Attempt at a Solution



by using that identity i got:

([tex]\stackrel{2n}{n}[/tex]) = (2n/n) ([tex]\stackrel{2n-1}{n-1}[/tex])
= (2) ([tex]\stackrel{2n-1}{n-1}[/tex])

i thought anything multiplied by 2 is an even number. but then again this is discrete math. how would i inductively show that this is true?
 
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  • #2
That's pretty much it. The definition of "even number" is that it is of the form 2k for some integer k. Do you already know that [itex]\left(\begin{array}{c}n\\2i\end{array}\right)[/itex] is always an integer?
 
  • #3
oh yeah! I forgot about that! thanks!
 

FAQ: Proving the Evenness of (\stackrel{2n}{n}) Using the Binomial Theorem

What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand a binomial expression raised to a positive integer power. It is expressed as (a + b)^n = Σ(n choose k) * a^(n-k) * b^k, where n is the power, a and b are the terms, and k is the index of summation.

How do I use the binomial theorem to expand a binomial expression?

To use the binomial theorem, you need to know the values of n, a, and b. Then, simply plug these values into the formula (a + b)^n = Σ(n choose k) * a^(n-k) * b^k and simplify the expression. Make sure to use the correct coefficients for each term.

Can the binomial theorem be used for negative or fractional powers?

No, the binomial theorem can only be used for positive integer powers. For negative or fractional powers, a different method, such as the Pascal's triangle or the general term formula, must be used to expand a binomial expression.

What are some real-life applications of the binomial theorem?

The binomial theorem has various applications in different fields of science and engineering, such as in probability and statistics, genetics, and physics. It is also used in the development of mathematical models for predicting patterns and outcomes in experiments and studies.

Are there any limitations to the binomial theorem?

Yes, the binomial theorem can only be used for expanding binomial expressions. It cannot be applied to expressions with more than two terms or expressions that do not follow the binomial form. Additionally, the binomial theorem assumes that the terms a and b are constants, and does not account for variables or changing values.

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