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Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$

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In summary, a binomial sum is a mathematical expression used in combinatorics and probability to calculate the number of possible combinations or outcomes. To calculate a binomial sum, you first need to identify the values of n and k and use the binomial coefficient formula. The 1/2^k term in the formula represents the probability of a successful outcome, and the formula is closely related to Pascal's triangle. The binomial sum formula can be used in various real-life situations, such as calculating probabilities and genetic outcomes.

- #1

- 247

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Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$

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= \sum_{k=0}^n \frac{1}{2^k} [z^n] (1+z)^{n+k}

= [z^n] (1+z)^n \sum_{k=0}^n \frac{1}{2^k} (1+z)^k

\\ = [z^n] (1+z)^n

\frac{1-(1+z)^{n+1}/2^{n+1}}{1-(1+z)/2}

= [z^n] (1+z)^n

\frac{2-(1+z)^{n+1}/2^{n}}{1-z}

\\ = 2\times 2^n

- [z^n] (1+z)^{2n+1} \frac{1}{2^n} \frac{1}{1-z}

\\ = 2\times 2^n

- \frac{1}{2^n} \sum_{k=0}^n {2n+1\choose k}

= 2\times 2^n - \frac{1}{2^n} \frac{1}{2} 2^{2n+1}

= 2^n.$$

A binomial sum is a mathematical expression that involves adding together terms of the form (n choose k) multiplied by a constant or variable. The binomial sum formula is commonly used in combinatorics and probability to calculate the number of possible combinations or outcomes.

To calculate a binomial sum, you first need to identify the values of n and k. Then, use the binomial coefficient formula (n choose k) = n! / (k!(n-k)!) to find the value of (n choose k). Finally, multiply this value by the constant or variable and add up all the terms to get the final result.

The 1/2^k term in the binomial sum formula represents the probability of a successful outcome in a binomial experiment. In other words, it is the probability of getting a specific combination of successes and failures in a series of trials. This term is often used when calculating the probability of events in statistics and probability theory.

The binomial sum formula is closely related to Pascal's triangle, a geometric arrangement of numbers where each number is the sum of the two numbers directly above it. The coefficients of the binomial sum can be found by looking at the corresponding row in Pascal's triangle and the exponents of the variables in the binomial sum correspond to the positions in the triangle.

The binomial sum formula can be used in a variety of real-life situations, such as calculating the probability of flipping a coin a certain number of times and getting a specific number of heads, or finding the number of possible combinations when selecting a certain number of items from a larger set. It is also used in the field of genetics to calculate the probability of specific genetic outcomes in offspring.

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