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\title{General Mathematics and Computational Science I}
\author{Midterm II}
\date{November 3, 2005}
\maketitle

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\begin{enumerate}

\item Cities A and B are connected as follows:

\begin{center}
\includegraphics[width=6cm]{cities.pdf}
\end{center}
  
\begin{enumerate}
\item In how many ways can you travel from A to B without taking any
road segment twice?  Explain.
\item In how many ways can you travel from A to B and back without
taking any road segment twice?  Explain.
\end{enumerate}
\hfill (5+5)

\item Prove, using generating functions or otherwise, that
\begin{equation*}
  \sum_{k=1}^n k \, \binom{n}{k} = n \, 2^{n-1} \,.
\end{equation*}
\hfill (10)


\item Four people get into an elevator in a six story building.  What
is the probability that they all get off at different floors?  (Assume
that each floor is equally likely to be visited.)
\hfill (10)

\item Solve the recurrence relation
\begin{equation*}
  a_{n+1} = 3 \, a_n - 2 \, a_{n-1}
\end{equation*}
with $a_0=3$ and $a_1=5$, using the method of generating functions. 
\hfill (10)

\item Show that, for a fixed perimeter, the rectangle with the
largest area is the square.
\mbox{\qquad}\hfill (10)

\item Let $a_1, \dots, a_n$ be positive.  Show that
\begin{equation*}
  \frac{a_1}{a_2} + \frac{a_2}{a_3} + \dots + \frac{a_n}{a_1} 
  \geq n \,. 
\end{equation*}

\end{enumerate}


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