\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\begin{document}

\title{General Mathematics and Computational Science I}
\author{Exercise 18}
\date{November 30, 2006}
\maketitle

\begin{enumerate}

\item Show that
\[
  \frac1{(1-x)^2} = 1 + 2 \, x + 3 \, x^2 + 4 \, x^3 + \dots \,.
\]

\emph{Hint:} Use the result of Exercise~9, Question~1.

\item Use the method of generating functions to solve the difference
equation
\begin{equation*}
  a_{n+1} = 2 \, a_{n} - a_{n-1}
\end{equation*}
with $a_0=1$ and $a_1=2$.

\emph{Hint:} Use the result from Question~1 above.


\item Find the equilibrium points of the system
\begin{gather*}
  x_{n+1} = 2 \, x_n - y_n \,, \\
  y_{n+1} = x_n + 3 \, y_n \,.
\end{gather*}
  


\end{enumerate}
\end{document}
