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\title{Math 480 (Spring 2007): Homework 7}
\author{\bf Due: Monday, May 14}
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\noindent{\bf There are 6 exciting problems.} Each problem is worth 6 points
and parts of multipart problems are worth equal amounts.  You may work
with other people and use a computer, unless otherwise stated.  Acknowledge
those who help you.\\

\begin{enumerate}

\item Find a continued fraction that equals each
of the following rational numbers:
\begin{enumerate}
\item $13/7$
\item $-9/13$
\item $21/13$
\end{enumerate}

\item Find the value (which is a rational number)
of each of the following continued fractions.
\begin{enumerate}
\item $[1,2,3]$
\item $[0,1,5,2]$
\item $[3,7,15]$
\end{enumerate}

\item Let $f_n$ be the $n$th Fibonacci number, so $f_1=1$, 
$f_2=1$, and for $n \geq 3$ we have $f_n = f_{n-1} + f_{n-2}$.
Prove that the continued fraction expansion of 
$f_{n+1}/f_{n}$ consists of $n$ $1$'s, i.e., 
$$
  \frac{f_{n+1}}{f_{n}} = [1,1,\ldots, 1].
$$

% page 257 of Kumanduri.
\item Prove that if $[a_0\ldots, a_n]$ and $[b_0,\ldots b_m]$
are two simple continued fractions that have the same value,
and that $a_i >0, b_j > 0$ for all $i,j$,
and $a_n > 1$ and $b_m > 1$, then $n=m$ and $a_i = b_i$ for
all $i$.  Thus the continued fraction expansion 
of a rational number is unique if the last term is 
required to be larger than $1$.

\item Show how to use continued fractions
to find a rational number $a/b$ in lowest terms such that 
$$\left|\frac{a}{b} - \sqrt[3]{2}\right| < \frac{1}{b^2} < 0.001.$$

\item The number $0.195876$ is a decimal approximation to a rational
number $a/b$ with $|b| < 100$.  Show how to use
continued fractions to find $a/b$.


\end{enumerate}

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