Answer: the ratios go getting closer and closer to the golden ratio.
If you extend the sequences:
A = 2,3,5,8,13,21, 34, 55, ...
B = 3,5,8,13,21,34,55,89
And the next ratios B/A are:
55/34 = 1.617647
89/55 = 1.6181818
The golden ratio is an irrational number, which means that it has infinite decimal digits not periodic.
Given a segment of length z = x + y, the golden ration is defined in this way: split the segment into two segments, one of length x and the other of length y. If x divided y is equal to z divided by x, then z / x is the golden ratio.
This is the expression, that represents the definition and permits to calculate it::
z x
Golden ratio, G = ---- = ---------
x y
Given that z = x + y =>
x + y x
G = ------- = -----
x y
x y
=> G = ----- + ---
x x
=> G = 1 + 1 / G
=> G^2 = G + 1
=> G^2 - G - 1 = 0
Using the quadratic formula you get:
1 +/- √ 1 - (4)(1)(-1) ] 1 +/- √5
G = ---------------------------- = ----------
2(1) 2
Taknig the positive solution: G = [1 + √5] / 2 = 1.618033988749....