If n≥8, and the coefficients of $\displaystyle x^7$ and $\displaystyle x^8$ in the expansion of $\displaystyle (3+x/2)^n$ 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?
If n≥8, and the coefficients of $\displaystyle x^7$ and $\displaystyle x^8$ in the expansion of $\displaystyle (3+x/2)^n$ 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?
$\displaystyle
\left( {3 + \frac{x}
{2}} \right)^n = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c}
n \\
k \\
\end{array} } \right)3^k \left( {\frac{x}
{2}} \right)^{n - k} }
$
now we equate the aforementioned coefficients:
$\displaystyle
\left( {\begin{array}{*{20}c}
n \\
{n - 7} \\
\end{array} } \right)3^{n - 7} \left( {\frac{1}
{2}} \right)^7 = \left( {\begin{array}{*{20}c}
n \\
{n - 8} \\
\end{array} } \right)3^{n - 8} \left( {\frac{1}
{2}} \right)^8
$
$\displaystyle \begin{gathered}
\Leftrightarrow 6\frac{{n!}}
{{7!\left( {n - 7} \right)!}} = \frac{{n!}}
{{8!\left( {n - 8} \right)!}} \hfill \\
\Leftrightarrow 48 = n - 7 \Rightarrow n = 55 \hfill \\
\end{gathered} $