Solve the quadratic equation x2 = 7x + 1. This equation is equivalent
to
which has solutions given by the infinite
continued fraction
This is because, if we think of this last equation as being
x = 7 + 1/y, then clearly y = x.
Show that the sequence of numbers
|
7 + |
1
7
|
, 7 + |
1
|
, 7 + |
1
|
, ¼ |
|
gives better and better approximations to one of the solutions of the
original quadratic equation. [Note that to find these approximations you can
simply repeat the steps: 'take reciprocal, add 7', over and over again].
Find integers a and b, with b less than 400, such that a/b ,
is equal to Ö53 correct to six significant figures.
Now consider x2 = 5x + 1, ...
[See the articles
Continued Fractions I and
Continued Fractions II. ]