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 fol lowing rational numbers:
(a) 13/7
(b) −9/13
(c) 21/13
2. Find the value (which is a rational number ) of each of the following
continued
fractions.
(a) [1, 2, 3]
(b) [0, 1, 5, 2]
(c) [3, 7, 15]
3. Let 
be the nth Fibonacci number, so
, and for n ≥ 3 we have
. Prove that the continued fraction expansion
of 
consists
of n 1’s, i.e.,
4. Prove that if 
and
are two simple continued fractions that
have the same value, and that 
for all i, j,
and 
and 
,
then n = m and 
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
6. 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.
