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\begin{document} %(
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\title{Problem Set 05}
% YOUR NAME GOES HERE under Author
\author{Your Name}
\maketitle

\begin{enumerate} %(
\item Write a general equation for the following sequences:  (For
example, a general equation for the sequence $2, 5, 10, 17, 26, \cdots$
would be $a_n = n^2 + 1$.  The equations should be closed form, and
they should be single equations.  For example  ``$a_n = \f{n}{2}$ if
$n$ is even, and $a_n = \f{n+1}{2}$ if $n$ is odd.'' wouldn't be
an acceptable answer for part c.)
\begin{enumerate} %( 
\item $3, 5, 7, 9, 11, \cdots$

% Answer below this line


\item $1, 5, 7, 17, 31, 65, 127, 257, \cdots$ (Hint, remember that
$\lx(-1\rx)^i = 1$ for even $i$, and $\lx(-1\rx)^i = -1$ for odd $i$.)

% Answer below this line


\item $1, 1, 2, 2, 3, 3, 4, 4, \cdots$  (BONUS points if you can do
this without using any ``rounding'' operators such as floor or ceiling.)

% Answer below this line


\end{enumerate} %)

\item Write a closed form formula for the following summations:  (For
example, a closed form formula for $\sum_{i = 1}^n 2i$ would be
$n\lx(n + 1\rx)$.)
\begin{enumerate} %(
\item $\sum_{i=1}^n \lx(2i + 3 + 9^i\rx)$

% Answer below this line


\item $\sum_{i=1}^n \lx(\lx(-1\rx)^i \cdot 9^i\rx)$

% Answer below this line


\item $\sum_{i=1}^n \sum_{j=1}^n ij$  

% Answer below this line


\item $\sum_{i=1}^n \sum_{j=1}^i j$ (Hint, $\sum_{i=1}^n i^2 = \f{n
\lx(n+1\rx) \lx(2n+1\rx)}{6}$.  Also note that the interior summation
is from $j = 1$ to $i$, not $n$.)

% Answer below this line


\end{enumerate} %)

\item Give a recursive definition for the following sequence:  (Be
sure to specify the base case(s).)
$1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, \cdots$

% Answer below this line


\item To be a member of the Sons of the American Revolution, you have
to have ``at least one ancestor who supported the cause of American
Independence during the years 1774-1783''.  Let's suppose that
everyone who fits this criterion is a member.

If we define $S$ to be the set of people who ``supported the cause of
American Independence during the years 1774-1783'', and we define
$p\lx(x, y\rx)$ to be ``$x$ is a parent of $y$'', then give a
recursive definition of $R$: the set of members of the Sons of the
American Revolution in terms of $S$ and $p$.

% Answer below this line


\item BONUS: The Fibonacci numbers are defined by the equation
\begin{displaymath} %(
f_n = f_{n-1} + f_{n-2}
\end{displaymath} %)
where $f_1 = f_2 = 1$.

Prove that, in the limit as $n \rightarrow \infty$,
$\f{f_{n+1}}{f_{n}} = \phi$, where $\phi$ is the Golden Ratio, given
by $\f{1+\sqrt{5}}{2}$.

Hint, you may want to prove that the following closed form equation
gives $f_n$:
\begin{displaymath} %(
f_n = \f{\lx(1+\sqrt{5}\rx)^n - \lx(1-\sqrt{5}\rx)^n}{2^n \sqrt{5}}
\end{displaymath} %)
then use this as a {\em lemma} in your proof.

% Answer below this line


\end{enumerate} %)
\end{document} %)