Question

Length B = 3, 5, 8, 13, 21, 34

Answer 1

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....