# Sum over k from 0 to an even n of abs(k/n - 1/2)*C(n,k)=(1/2)*C(n,n/2)

• neginf
In summary, the conversation discusses the concept of odd numbers and how they have a middle term equal to 0. It also explores the relationship between C(n,k) and C(n,n-k), and how the sum can be simplified using these equations. The conversation also mentions specific examples and formulas that can be useful in understanding even numbers.
neginf
Odd number of terms so there is a middle term. It is equal to 0.
Since C(n,k)=C(n,n-k), abs(k/n - 1/2)=abs([n-k]/n - 1/2), and what's in the abs() increases and is positive k goes from 0 to n/2 -1,
the sum is twice the sum of the first n/2 - 1 terms, 2*(1/2 - k/n)*C(n,k).
Added C(n,k) from 0 to n, C(n-1,k-1) from 1 to n-1 seperately. Still do not see how to get.
Specific examples are equal to (1/2)*C(n,n/2) but how can this be shown for any even n ?

Theese two formulas should be useful:
$\sum_{k=0}^nC(n,k)=2^n$
$\sum_{k=0}^nkC(n,k)=n2^{n-1}$
Does it help?

## 1. What is the meaning of "Sum over k from 0 to an even n of abs(k/n - 1/2)*C(n,k)=(1/2)*C(n,n/2)"?

This equation represents a mathematical relationship between two sets of numbers, where one set is the sum of values from 0 to an even number n, and the other set is the combination of n items taken k at a time. The left side of the equation measures the difference between each value in the first set and half of n, while the right side calculates the combination of n items taken halfway. The equation states that these two sets of numbers are equal.

## 2. How is this equation relevant to scientific research?

This equation is relevant in many areas of scientific research, particularly in statistics and probability. It can be used to calculate the probability of certain outcomes in experiments or to analyze data sets. It also has applications in fields like genetics, where it can be used to calculate the likelihood of certain genetic traits being inherited.

## 3. What is the significance of using an even value for n in this equation?

The even value for n is significant because it ensures that the sum on the left side of the equation includes both positive and negative values, resulting in a cancellation effect. This allows for a more precise calculation of the combination on the right side of the equation, leading to a more accurate result.

## 4. Can this equation be simplified or rewritten in a different form?

Yes, this equation can be simplified using mathematical properties and identities. For example, the absolute value and the combination can be expanded and then simplified, resulting in a shorter form of the equation. It can also be rewritten in terms of factorials or binomial coefficients.

## 5. How can this equation be applied in real-world scenarios?

This equation can be applied in various real-world scenarios, such as in statistical analysis, genetics, and finance. It can be used to calculate the probability of certain outcomes, analyze data sets, or make predictions. In genetics, it can be used to determine the likelihood of certain genetic traits being passed down through generations. In finance, it can be used to calculate the expected value of an investment or to analyze risk and return.

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