Math 480 (Spring 2007): Homework 7

Due: Monday, May 14

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.

1. Find a continued fraction that equals each of the following rational numbers:
   1. $13/7$
   2. $-9/13$
   3. $21/13$

2. Find the value (which is a rational number) of each of the following continued fractions.
   1. $[1,2,3]$
   2. $[0,1,5,2]$
   3. $[3,7,15]$

3. 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.,
   $$\frac{f_{n+1}}{f_{n}} = [1,1,\ldots, 1].$$

4. 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 $1$ .

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

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