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Math Help - Binomial theorem question

  1. #1
    Newbie
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    Binomial theorem question

    If n≥8, and the coefficients of x^7 and x^8 in the expansion of (3+x/2)^n are equal, what is n?


    I equated the coefficents, but end up with n being 8.03 something, which can't be right.

    Any help?
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  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
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    <br />
\left( {3 + \frac{x}<br />
{2}} \right)^n  = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   k  \\<br /> <br />
 \end{array} } \right)3^k \left( {\frac{x}<br />
{2}} \right)^{n - k} } <br />

    now we equate the aforementioned coefficients:

    <br />
\left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   {n - 7}  \\<br /> <br />
 \end{array} } \right)3^{n - 7} \left( {\frac{1}<br />
{2}} \right)^7  = \left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   {n - 8}  \\<br /> <br />
 \end{array} } \right)3^{n - 8} \left( {\frac{1}<br />
{2}} \right)^8 <br />


    \begin{gathered}<br />
   \Leftrightarrow 6\frac{{n!}}<br />
{{7!\left( {n - 7} \right)!}} = \frac{{n!}}<br />
{{8!\left( {n - 8} \right)!}} \hfill \\<br />
   \Leftrightarrow 48 = n - 7 \Rightarrow n = 55 \hfill \\ <br />
\end{gathered}
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