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

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```							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
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14 15 16 -36

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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

```
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