Use of binomial theorem in a sum of binomial coefficients?

  • #1
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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?
 

Answers and Replies

  • #2
Ray Vickson
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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?

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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