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.