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

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1. Oct 25, 2015

### gruba

1. The problem statement, all variables and given/known data
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)?

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

3. 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?

2. Oct 25, 2015

### Ray Vickson

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?
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.