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\title{General Mathematics and Computational Science I}
\author{Exercise 8}
\date{October 19, 2006}
\maketitle

\begin{enumerate}

\item Show that
\begin{equation*}
  \sum_{k=0}^n\binom{n+k}{k}\frac{1}{2^k}
  = 2^n
\end{equation*}
\textbf{Hint:} Denote the left hand side by $f(n)$ and prove that
$f(n+1)=2 \, f(n)$. Use Exercise 6, Problem 2.

\item Show that
\begin{equation*}
  \binom{n+1}{k+1}
  = \sum_{j=k}^n \binom{j}{k}
\end{equation*}
for every $0 \leq k \leq n$.

\item Use the method of generating functions to find a closed form
expression for the members of the generalized Fibonacci sequence
\begin{gather*}
  a_0 = A \,, \\
  a_1 = B \,, \\
  a_n = a_{n-1} + a_{n-2} \,.
\end{gather*}

\end{enumerate}
\end{document}

\item Use the method of generating functions to find a closed form
expression for the members of the sequence
\begin{gather*}
  b_0 = 1 \,, \\
  b_1 = \cos \theta \,, \\
  b_n = 2 \cos \theta \, b_{n-1} - b_{n-2} \,,
\end{gather*}
where $\theta$ is an arbitrary real number.