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Use of binomial theorem in a sum of binomial coefficients?

  1. Oct 25, 2015 #1
    1. The problem statement, all variables and given/known data
    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)?

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

    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. jcsd
  3. Oct 25, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    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
    [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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