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\title{General Mathematics and Computational Science I}
\author{Midterm II}
\date{November 7, 2006}
\maketitle

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\begin{enumerate}

\item

In chemistry, benzene is a molecule made of six carbon atoms forming a
hexagon, each of which also bonds to a hydrogen atom (H).  In how many
ways can hydrogen atoms be substituted by two methyl groups (M) and
one ethyl group (E) as to form distinct molecules? 

Three possible configurations are shown as examples: 
\begin{center}
\includegraphics[width=12cm]{benzene.eps}
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Note that the left two examples are distinct molecules, while the
right two are a pair of identical molecules!  

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\item Prove the so-called \emph{Bernoulli inequality}
\begin{gather*}
  (1+x)^n \geq 1 + n \, x
\end{gather*}
for any real number $x>-1$ and $n \in \mathbb{N}$. 
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\item
\begin{enumerate}
\item You flip a coin five times in a row.  What is the probability
that it comes up heads five times?

\item You flip a coin ten times in a row.  Which of the following
outcomes is more likely?  HHHHHHHHHH or THTTHHTHHT?  Explain!
\end{enumerate}
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\item Use the method of generating functions to find a closed form
expression for the members of the sequence 
\begin{equation*}
  a_{n+1} = 2 \, a_{n-1} - a_n
\end{equation*}
where $a_0=1$ and $a_1=-2$.
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\item Find the largest value of the function $f(x,y)=xy$ on the set of
points satisfying the inequality $x^2+xy+y^2\leq 1$. 
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\item Show that
\begin{equation*}
  (a+b+c)\, \biggl( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \biggr)\geq 9 
\end{equation*}
for arbitrary positive real numbers $a,b$ and $c$. When does
equality hold?
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\end{enumerate}


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