Fibonacci Sequences:

1
1
2
3
5
8
13
21
34
55
89
144
233
377
610

Here Is the fibonacci sequence. Notice that each term is the sum of the previous two terms.

1
1	1
2	2
3	1.5
5	1.66666666666667
8	1.6
13	1.625
21	1.61538461538462
34	1.61904761904762
55	1.61764705882353
89	1.61818181818182
144	1.61797752808989
233	1.61805555555556
377	1.61802575107296
610	1.61803713527851

In this chart, the ratio of each two previous terms is calculated. Notice that it approaches a fixed number 1.618...This number is called phi.

1
1	1
2	2	2
3	1.5	3
5	1.66666666666667	2.5
8	1.6	2.66666666666667
13	1.625	2.6
21	1.61538461538462	2.625
34	1.61904761904762	2.61538461538462
55	1.61764705882353	2.61904761904762
89	1.61818181818182	2.61764705882353
144	1.61797752808989	2.61818181818182
233	1.61805555555556	2.61797752808989
377	1.61802575107296	2.61805555555556
610	1.61803713527851	2.61802575107296

This chart shows the ratio of each term to the term previous to the one before it. Notice that this ratio also approaches a fixed number. In this case the number is 2.618...or phi + 1. This is the square of phi.

Here I changed the initial values. f(0) = 1 and f(1) = 3. This is called the Lucas sequence.

1
3	3
4	1.33333333333333	4
7	1.75	2.33333333333333
11	1.57142857142857	2.75
18	1.63636363636364	2.57142857142857
29	1.61111111111111	2.63636363636364
47	1.62068965517241	2.61111111111111
76	1.61702127659574	2.62068965517241
123	1.61842105263158	2.61702127659574
199	1.61788617886179	2.61842105263158
322	1.61809045226131	2.61788617886179
521	1.61801242236025	2.61809045226131
843	1.61804222648752	2.61801242236025
1364	1.61803084223013	2.61804222648752

The first ratio still approaches phi.

Notice that the same relationship exists for the second ratio as well.

Now I am going to change f(0) = 2 and f(1) = 10. These are just arbitrary numbers that I picked.

2
10	5
12	1.2	6
22	1.83333333333333	2.2
34	1.54545454545455	2.83333333333333
56	1.64705882352941	2.54545454545455
90	1.60714285714286	2.64705882352941
146	1.62222222222222	2.60714285714286
236	1.61643835616438	2.62222222222222
382	1.61864406779661	2.61643835616438
618	1.61780104712042	2.61864406779661
1000	1.61812297734628	2.61780104712042
1618	1.618	2.61812297734628
2618	1.61804697156984	2.618
4236	1.61802902979374	2.61804697156984

Again, the same relationship exists!

To see more on Fibonacci sequences, click HERE.

To see more on the Lucas Sequence, click HERE.

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