This can be shown if you use the closed form formula known as the Binet formula for the nth term of a Fibonacci sequence

							Convergence Of Ratios In Fibonacci Sequences
Rod Rodrigues
Ratios in Fibonacci Sequences                                                            u1:        12
Select two starting numbers and inspect the successive ratios                            u2:        -4
                                                                                                     48
Terms     ui/u(i+1)   ui/u(i-1)    Diff                                                              47
     12                                                                                              46
     -4            -3 -0.333333     2.6667                     1           1
                                                                Ratios in Fibonacci Sequences 45
      8          -0.5         -2       -1.5                    1             1                       44
                                              4
      4             2        0.5       -1.5                    1             1                       43
     12    0.3333333           3    2.6667    3
                                                               1             1                       42
     16          0.75 1.333333      0.5833                     1             1                       41
     28    0.5714286        1.75    1.1786    2                1             1                       40
     44    0.6363636 1.571429       0.9351                     1             1                       39
                                              1
     72    0.6111111 1.636364       1.0253                     1             1                       38
    116    0.6206897 1.611111       0.9904    0                1             1                       37
    188    0.6170213    1.62069     1.0037         1   2   3   14   5   6   71   8   9                14
                                                                                         10 11 12 13 36

    304    0.6184211 1.617021       0.9986    -1               1             1                       35
    492    0.6178862 1.618421       1.0005                     1             1                       34
                                              -2
    796    0.6180905 1.617886       0.9998                     1             1                       33
   1288    0.6180124    1.61809     1.0001    -3               1             1                       32
   2084    0.6180422 1.618012             1                    1             1                       31
                                              -4
   3372    0.6180308 1.618042             1                    1             1                       30
   5456    0.6180352 1.618031             1                    1             1                       29
                                                                                                     28
                 17         17                                                                       27
Terms for convergence to 6 decimal places                                                            26
                                                                                                     25
                                                                                                     24
                                                                                                     23
                                                                                                     22
                                                                                                     21
                                                                                                     20
                                                                                                     19
                                                                                                     18
                                                                                                     17
                                                                                                     16
                                                                                                     15
                                                                                                     14
                                                                                                     13
                                                                                                     12
                                                                                                     11
                                                                                                     10
                                                                                                      9
                                                                                                      8
  7
  6
  5
  4
  3
  2
  1
 -1
 -2
 -3
 -4
 -5
 -6
 -7
 -8
 -9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22
-23
-24
-25
-26
-27
-28
-29
-30
-31
-32
-33
-34
-35
-36
-37
-38
-39
-40
-41
-42
-43
-44
-45
-46
-47
-48
-49
-50
         -48
         -47
         -46
         -45
         -44
         -43
         -42
         -41
         -40
         -39
         -38
         -37
          17
14 15 16 -36

         -35
         -34
         -33
         -32
         -31
         -30
         -29
         -28
         -27
         -26
         -25
         -24
         -23
         -22
         -21
         -20
         -19
         -18
         -17
         -16
         -15
         -14
         -13
         -12
         -11
         -10
          -9
          -8
-7
-6
-5
-4
-3
-2
-1
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Convergence of Fibonacci ratios to the Golden Ratios to 6 decimal places occurs by the 17th ratio.
This can be shown if you use the closed form formula known as the Binet formula for the nth term
of a Fibonacci sequence.
See                     http://www.rodsweb.org/phi/fib1.htm
The third column shows when the DIFFERENCE converges to 1.
The difference between the two exact number is exactly 1.



              5 1       5 1
                 , 
               2          2
                 5  1 5 1      5 1  5 1 1 1
                                  1
                  2        2    2 2 2 2 2 2

So it is also of interest to see how long it takes for the difference to converge to 1.
This sheet counts the numebr of terms for the ratios to converge to the GoldenMean
to six decimal places.
by the 17th ratio.
 for the nth term