Results 1 to 2 of 2

Math Help - Binomial theorem

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    8

    Binomial theorem

    Using the binomial theorem, write down expressions for (1+x)^n , (1-x)^n , (1-{x^2})^n. By comparing terms in x^{2r}, prove that for any r with 0≤r≤n

    {\left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   r  \\<br /> <br />
 \end{array} } \right)} = \sum\limits_{k = 0}^{2r}(-1)^{r-k} {\left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   k  \\<br /> <br />
 \end{array} } \right) }<br />
{\left({\begin{array}{*{20}c}<br />
n \\<br />
2r - k \\<br />
\end{array} } \right) }



    Got expansions for the three binomial expansions, but I don't know how to compare the terms in x^{2r}

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by smwatson View Post
    Using the binomial theorem, write down expressions for (1+x)^n , (1-x)^n , (1-{x^2})^n. By comparing terms in x^{2r}, prove that for any r with 0≤r≤n

    {\left( {\begin{array}{*{20}c}<br />
n \\<br />
r \\<br /> <br />
\end{array} } \right)} = \sum\limits_{k = 0}^{2r}(-1)^{r-k} {\left( {\begin{array}{*{20}c}<br />
n \\<br />
k \\<br /> <br />
\end{array} } \right) }<br />
{\left({\begin{array}{*{20}c}<br />
n \\<br />
2r - k \\<br />
\end{array} } \right) }



    Got expansions for the three binomial expansions, but I don't know how to compare the terms in x^{2r}
    Note that (1+x)^n (1-x)^n = (1-{x^2})^n. So multiply the two expansions on the left hand side and compare the resulting coefficient of x^{2r} with the coefficient of x^{2r} in the expansion on the right hand side.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. binomial theorem
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 20th 2010, 02:12 PM
  2. Binomial Theorem or Binomial Coefficient
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: October 2nd 2009, 01:06 PM
  3. Binomial Theorem?
    Posted in the Algebra Forum
    Replies: 4
    Last Post: April 24th 2009, 08:52 AM
  4. Binomial Theorem
    Posted in the Algebra Forum
    Replies: 3
    Last Post: January 5th 2009, 02:31 AM
  5. Binomial Theorem
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: January 1st 2009, 01:24 PM

Search Tags


/mathhelpforum @mathhelpforum