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\title{General Mathematics and Computational Science I}
\author{Exercise 21}
\date{November 29, 2007}
\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~11, Question~2.

\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 Solve the difference equation from Question~2 using the theorem
proved in class (Theorem 6 in the class notes).


\end{enumerate}
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