Use of binomial theorem in a sum of binomial coefficients?

In summary, the binomial theorem can be used to represent sums with binomial coefficients, but the process of finding these representations can be difficult and is best done with the help of a computer.
  • #1
gruba
206
1

Homework Statement


How to use binomial theorem for finding sums with binomial coefficients?
Example: [tex]S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-...[/tex]

How to represent this sum using [itex]\sum\limits[/itex] notation (with binomial theorem)?

Homework Equations


[itex](a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k[/itex]

The Attempt at a Solution


I am completely stuck on this types of problems (modify binomial formula so that it gives some arbitrary sum). Could someone please explain this example?
 
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  • #2
gruba said:

Homework Statement


How to use binomial theorem for finding sums with binomial coefficients?
Example: [tex]S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-...[/tex]

How to represent this sum using [itex]\sum\limits[/itex] notation (with binomial theorem)?

Homework Equations


[itex](a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k[/itex]

The Attempt at a Solution


I am completely stuck on this types of problems (modify binomial formula so that it gives some arbitrary sum). Could someone please explain this example?

Are the coefficients supposed to be ##(-3)^0, (-3)^1, (-3)^2, \ldots##? If so, what is preventing you from using the formula you wrote under heading 2?
gruba said:

Homework Statement


How to use binomial theorem for finding sums with binomial coefficients?
Example: [tex]S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-...[/tex]

How to represent this sum using [itex]\sum\limits[/itex] notation (with binomial theorem)?

Homework Equations


[itex](a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k[/itex]

The Attempt at a Solution


I am completely stuck on this types of problems (modify binomial formula so that it gives some arbitrary sum). Could someone please explain this example?

I assume you mean
[tex] S_n = \begin{cases}
\displaystyle \sum_{k=0}^{m-1} (-3)^k {2m \choose 2k+1}, & n = 2m \\
\displaystyle \sum_{k=0}^m (-3)^k {2m+1 \choose 2k+1}, & n = 2m + 1
\end{cases}
[/tex]
If so, these are not particularly easy to determine. A computer algebra package such as Maple or Mathematica can determine the answers. You can also submit them to the free on-line package Wolfram Alpha.
 
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1. What is the binomial theorem and how is it used in a sum of binomial coefficients?

The binomial theorem is a mathematical formula that expresses the expansion of a binomial raised to a positive integer power. It states that the sum of the coefficients of the terms in the expansion is equal to the binomial coefficient of that power. This theorem is used to simplify and solve problems involving binomial coefficients, which are the coefficients of the terms in the expanded form of a binomial.

2. How do you calculate the binomial coefficient in a sum of binomial coefficients?

The binomial coefficient is calculated using the formula n choose r, where n represents the total number of items and r represents the number of items chosen at a time. This is also known as a combination, and the formula is n! / (r!(n-r)!), where ! represents factorial. For example, if we have 5 items and we want to choose 2 at a time, the binomial coefficient would be 5 choose 2, or 5! / (2!(5-2)!), which simplifies to 10.

3. How is the binomial theorem used to expand a binomial raised to a power?

The binomial theorem can be used to expand a binomial raised to a power by using the formula (a + b)^n = n choose 0 * a^n + n choose 1 * a^(n-1) * b + n choose 2 * a^(n-2) * b^2 + ... + n choose n * b^n, where a and b are the terms of the binomial and n is the power. This formula can be used to find the coefficients of each term in the expansion.

4. Can the binomial theorem be used for non-integer powers?

No, the binomial theorem can only be used for positive integer powers. This is because the binomial coefficients are only defined for positive integers. However, the binomial theorem can be extended to include fractional powers by using the generalized binomial theorem, which involves using binomial coefficients with negative and non-integer values.

5. How is the binomial theorem used in real life applications?

The binomial theorem has various applications in fields such as statistics, physics, and engineering. It can be used to solve problems involving combinations and probability, as well as in the expansion of polynomial equations. It is also used in the field of genetics to calculate the chances of inheriting certain traits. Additionally, the binomial theorem is used in the development of algorithms and in computer programming.

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