For this assignment, I have chose to investigate the Fibbonnaci sequence. I had never heard of the Fibbonnaci sequence until my third year of college, so there is a lot I do not know. At this point all I do know is that it is generated by f(n) = f(n-1) + f(n-2). I will begin by generating the Fibbonnaci sequence on a spreadsheet. I will generate it up to f(25) and such that f(0) = 1 and f(1)= 1.
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 |
89 |
144 |
233 |
377 |
610 |
987 |
1597 |
2584 |
4181 |
6765 |
10946 |
17711 |
28657 |
46368 |
75025 |
Now that we can see how the Fibbonnaci sequence is constructed and what it looks like, we can begin to investigate it. I will start by taking the ratio of the adjacent terms or more formally g(n) = f(n-1)/f(n-2).
1 |
|
1 |
1 |
2 |
2 |
3 |
1.5 |
5 |
1.666666667 |
8 |
1.6 |
13 |
1.625 |
21 |
1.615384615 |
34 |
1.619047619 |
55 |
1.617647059 |
89 |
1.618181818 |
144 |
1.617977528 |
233 |
1.618055556 |
377 |
1.618025751 |
610 |
1.618037135 |
987 |
1.618032787 |
1597 |
1.618034448 |
2584 |
1.618033813 |
4181 |
1.618034056 |
6765 |
1.618033963 |
10946 |
1.618033999 |
17711 |
1.618033985 |
28657 |
1.61803399 |
46368 |
1.618033988 |
75025 |
1.618033989 |
As we can see, as f(n) gets larger g(n) approaches 1.618, which is the golden ratio. This was not too surprising to me because I had seen it before, I just did not remember it. Next, I start playing around with the f(0) and f(1) terms. I changed them to one and three, as the assignment recommended. I then took the ratio of its adjacent terms.
1 |
|
3 |
3 |
4 |
1.333333 |
7 |
1.75 |
11 |
1.571429 |
18 |
1.636364 |
29 |
1.611111 |
47 |
1.62069 |
76 |
1.617021 |
123 |
1.618421 |
199 |
1.617886 |
322 |
1.61809 |
521 |
1.618012 |
843 |
1.618042 |
1364 |
1.618031 |
2207 |
1.618035 |
3571 |
1.618034 |
5778 |
1.618034 |
9349 |
1.618034 |
15127 |
1.618034 |
24476 |
1.618034 |
39603 |
1.618034 |
64079 |
1.618034 |
103682 |
1.618034 |
167761 |
1.618034 |
This surprised me somewhat, but not too much because I figured it was a special case. I then began to plug in various numbers for f(0) and f(1). To my surprise, the golden ratio came out every time. Here are a couple of examples…
F(0) = 1 and f(1) = 3
1 |
|
3 |
3 |
4 |
1.333333 |
7 |
1.75 |
11 |
1.571429 |
18 |
1.636364 |
29 |
1.611111 |
47 |
1.62069 |
76 |
1.617021 |
123 |
1.618421 |
199 |
1.617886 |
322 |
1.61809 |
521 |
1.618012 |
843 |
1.618042 |
1364 |
1.618031 |
2207 |
1.618035 |
3571 |
1.618034 |
5778 |
1.618034 |
9349 |
1.618034 |
15127 |
1.618034 |
24476 |
1.618034 |
39603 |
1.618034 |
64079 |
1.618034 |
103682 |
1.618034 |
167761 |
1.618034 |
F(0) = 0 and f(1) = 2
0 |
|
2 |
|
2 |
1 |
4 |
2 |
6 |
1.5 |
10 |
1.666667 |
16 |
1.6 |
26 |
1.625 |
42 |
1.615385 |
68 |
1.619048 |
110 |
1.617647 |
178 |
1.618182 |
288 |
1.617978 |
466 |
1.618056 |
754 |
1.618026 |
1220 |
1.618037 |
1974 |
1.618033 |
3194 |
1.618034 |
5168 |
1.618034 |
8362 |
1.618034 |
13530 |
1.618034 |
21892 |
1.618034 |
35422 |
1.618034 |
57314 |
1.618034 |
92736 |
1.618034 |
F(0) = 300 and f(1) = 1
300 |
|
1 |
0.003333 |
301 |
301 |
302 |
1.003322 |
603 |
1.996689 |
905 |
1.500829 |
1508 |
1.666298 |
2413 |
1.600133 |
3921 |
1.624948 |
6334 |
1.615404 |
10255 |
1.61904 |
16589 |
1.61765 |
26844 |
1.618181 |
43433 |
1.617978 |
70277 |
1.618055 |
113710 |
1.618026 |
183987 |
1.618037 |
297697 |
1.618033 |
481684 |
1.618034 |
779381 |
1.618034 |
1261065 |
1.618034 |
2040446 |
1.618034 |
3301511 |
1.618034 |
5341957 |
1.618034 |
8643468 |
1.618034 |
F(0) = -6 and f(1) = -2
-6 |
|
-2 |
0.333333 |
-8 |
4 |
-10 |
1.25 |
-18 |
1.8 |
-28 |
1.555556 |
-46 |
1.642857 |
-74 |
1.608696 |
-120 |
1.621622 |
-194 |
1.616667 |
-314 |
1.618557 |
-508 |
1.617834 |
-822 |
1.61811 |
-1330 |
1.618005 |
-2152 |
1.618045 |
-3482 |
1.61803 |
-5634 |
1.618036 |
-9116 |
1.618033 |
-14750 |
1.618034 |
-23866 |
1.618034 |
-38616 |
1.618034 |
-62482 |
1.618034 |
-101098 |
1.618034 |
-163580 |
1.618034 |
-264678 |
1.618034 |
Now I will try taking the ratio of every other term or h(n) = f(n-1)/f(n-3). I have no idea what the ratio will be.
F(0) = 1 and f(1) = 1
1 |
|
1 |
|
2 |
2 |
3 |
3 |
5 |
2.5 |
8 |
2.666667 |
13 |
2.6 |
21 |
2.625 |
34 |
2.615385 |
55 |
2.619048 |
89 |
2.617647 |
144 |
2.618182 |
233 |
2.617978 |
377 |
2.618056 |
610 |
2.618026 |
987 |
2.618037 |
1597 |
2.618033 |
2584 |
2.618034 |
4181 |
2.618034 |
6765 |
2.618034 |
10946 |
2.618034 |
17711 |
2.618034 |
28657 |
2.618034 |
46368 |
2.618034 |
75025 |
2.618034 |
As you can see, the ratio seems to be the golden ratio plus one. Now I am going to check to see if this works for other values of f(0) and f(1).
F(0) = 0 and f(1) = 2
0 |
|
2 |
|
2 |
#DIV/0! |
4 |
2 |
6 |
3 |
10 |
2.5 |
16 |
2.666667 |
26 |
2.6 |
42 |
2.625 |
68 |
2.615385 |
110 |
2.619048 |
178 |
2.617647 |
288 |
2.618182 |
466 |
2.617978 |
754 |
2.618056 |
1220 |
2.618026 |
1974 |
2.618037 |
3194 |
2.618033 |
5168 |
2.618034 |
8362 |
2.618034 |
13530 |
2.618034 |
21892 |
2.618034 |
35422 |
2.618034 |
57314 |
2.618034 |
92736 |
2.618034 |
F(0) = -7 and f(1) = -99
-7 |
|
-99 |
|
-106 |
15.14286 |
-205 |
2.070707 |
-311 |
2.933962 |
-516 |
2.517073 |
-827 |
2.659164 |
-1343 |
2.602713 |
-2170 |
2.623942 |
-3513 |
2.615786 |
-5683 |
2.618894 |
-9196 |
2.617706 |
-14879 |
2.618159 |
-24075 |
2.617986 |
-38954 |
2.618052 |
-63029 |
2.618027 |
-101983 |
2.618037 |
-165012 |
2.618033 |
-266995 |
2.618034 |
-432007 |
2.618034 |
-699002 |
2.618034 |
-1131009 |
2.618034 |
-1830011 |
2.618034 |
-2961020 |
2.618034 |
-4791031 |
2.618034 |
F(0) = .001 and f(1) = .35
0.35 |
|
0.001 |
|
0.351 |
1.002857 |
0.352 |
352 |
0.703 |
2.002849 |
1.055 |
2.997159 |
1.758 |
2.500711 |
2.813 |
2.666351 |
4.571 |
2.600114 |
7.384 |
2.624956 |
11.955 |
2.615401 |
19.339 |
2.619041 |
31.294 |
2.61765 |
50.633 |
2.618181 |
81.927 |
2.617978 |
132.56 |
2.618055 |
214.487 |
2.618026 |
347.047 |
2.618037 |
561.534 |
2.618033 |
908.581 |
2.618034 |
1470.115 |
2.618034 |
2378.696 |
2.618034 |
3848.811 |
2.618034 |
6227.507 |
2.618034 |
10076.32 |
2.618034 |
F(0) = -654 and f(1) = 1.2
-654 |
|
1.2 |
|
-652.8 |
0.998165 |
-651.6 |
-543 |
-1304.4 |
1.998162 |
-1956 |
3.001842 |
-3260.4 |
2.49954 |
-5216.4 |
2.666871 |
-8476.8 |
2.599926 |
-13693.2 |
2.625029 |
-22170 |
2.615374 |
-35863.2 |
2.619052 |
-58033.2 |
2.617645 |
-93896.4 |
2.618182 |
-151930 |
2.617977 |
-245826 |
2.618056 |
-397756 |
2.618026 |
-643582 |
2.618037 |
-1041337 |
2.618033 |
-1684919 |
2.618034 |
-2726256 |
2.618034 |
-4411175 |
2.618034 |
-7137431 |
2.618034 |
-1.2E+07 |
2.618034 |
-1.9E+07 |
2.618034 |
I tried to pick the most random values for f(0) and f(1) and no matter what their value was the ratio was always the gold ratio plus one.
My final conclusions for this assignment are…
F(n) = f(n-1) + f(n-2) => f(n-1)/f(n-2) = 1.61… = (1 + sqrt(5)) / 2 = the golden ratio.
F(n) = f(n-1) + f(n-2) => f(n-1)/f(n-3) = 2.61… =[(1 + sqrt(5)) / 2] + 1 = the golden ratio + 1.