British Mathematical Olympiad 2004/5 - Arbelos

www.arbelos.co.uk/papers.html c○ Andrew Jobbings 2004
which gives a closed form for [a 1 , a 2 , . . . , a n ] in terms of f and g.
It follows that any rational number in the interval 0 < x < 1 belongs to S , since any such
rational may be expressed as a continued fraction of the form [a 1 , a 2 , . . . , a n ].
Example
For example,
giving the sequence of fractions
1
2
g
↦−→ 1 3
g
↦−→ 1 4
28
39
g
↦−→ 1 5
= [1, 2, 1, 1, 5]
= f ◦ g f ◦ f ◦ f ◦ g ( )
3 1
2
f
↦−→ 5 6
f
↦−→ 6 11
f
↦−→ 11
17
g
↦−→ 11
28
f
↦−→ 28
39 .
2