# Use of binomial theorem in a sum of binomial coefficients?

## Homework Statement

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

How to represent this sum using $\sum\limits$ notation (with binomial theorem)?

## Homework Equations

$(a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k$

## 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?

Ray Vickson
Homework Helper
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## Homework Statement

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

How to represent this sum using $\sum\limits$ notation (with binomial theorem)?

## Homework Equations

$(a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k$

## 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?

## Homework Statement

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

How to represent this sum using $\sum\limits$ notation (with binomial theorem)?

## Homework Equations

$(a+b)^n=\sum\limits_{k=0}^{n}{n\choose k}a^{n-k}b^k$

## 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
$$S_n = \begin{cases} \displaystyle \sum_{k=0}^{m-1} (-3)^k {2m \choose 2k+1}, & n = 2m \\ \displaystyle \sum_{k=0}^m (-3)^k {2m+1 \choose 2k+1}, & n = 2m + 1 \end{cases}$$
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.

• gruba