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\begin{document}


\begin{enumerate}
\item The \textbf{Fibonacci sequence}, $F_{n}$ is $1,1,2,3,5,8,13,21,...$ \
\ Every element is obtained by adding the previous two elements. The
recursive definition is%
\begin{eqnarray*}
F_{1} &=&1 \\
F_{2} &=&1 \\
F_{n+2} &=&F_{n}+F_{n+1}\text{ \ \ for all natural number }n\text{, with }%
n\geq 1
\end{eqnarray*}

Consider now another sequence, $q_{n}$ that is formed by taking the ratios
of consecutive elements of the Fibonacci sequence.%
\begin{equation*}
q_{n}=\dfrac{F_{n}}{F_{n+1}}\text{ \ \ for all natural number }n\text{.}
\end{equation*}%
Compute the decimal presentation of the first few elements of $q_{n}$. What
behavior do you notice?

\item Prove that consecutive elements of the Fibonacci sequence are
relatively prime.

\item Leo the 58th is a king in a country. Amazed by his own greatness, he
announces that the use of all currency is illegal, except for the use of $%
\$~5$ bills and $\$~8$ bills.

\begin{enumerate}
\item We see a product in a store that costs $\$~7$. Can we purchase it with
the legal currencies?

\item Is there a price that would be impossible to pay?

\item What would be the case if his son, Leo the 59th started to rule?

\item Ask further questions!
\end{enumerate}

\item Definition: (All letters denote positive integers) A number $n$ is 
\textbf{irreducible} if for all $a$ and $b$,%
\begin{equation*}
n=ab~\Longrightarrow ~a=1\text{ \ or  }b=1
\end{equation*}%
Definition: A number $n$ is \textbf{prime} if for all $a$ and $b$%
\begin{equation*}
n|ab~\Longrightarrow ~n|a\text{ \ or \ }n|b
\end{equation*}

\begin{enumerate}
\item Prove that if a number is prime, then it is irreducible.

\item Prove that if a number is irreducible, then it is a prime.
\end{enumerate}
\end{enumerate}

\end{document}