Here is a sequence of continued fractions:
X1=1,     X2 = 1
1+1
 ,     X3 = 1
1+ 1
1+ 1
 ,     X4 = 1
1+ 1
1+ 1
1 + 1
 ,...
Notice that
Xn+1 = 1
1 + Xn
 .
Now suppose that this sequence tends to a limit L as n® ¥ then put Xn+1=Xn=L and prove that
L = f- 1 = 1
f

where f is the Golden Ratio, the positive solution to the equation x2 - x -1 = 0.

Prove that
Xn= Fn
Fn+1
where Fn is a Fibonacci number from the sequence defined by the relation Fn+2=Fn+1+Fn where F1=1 and F2=1.

Hence show that the ratio of successive terms of the Fibonacci sequence
Fn+1
Fn
tends to the Golden Ratio as n® ¥.