An Exploration of the Fibonacci Sequence

Kenneth E. Montgomery

Fibonacci sequences are sequences of whole numbers generated by Equation 1:

Equation 1: , for all

The Fibonacci Sequence is generated by Equation 1, where. A spreadsheet was used to investigate this sequence and the ratios of its terms (Table 1).

n	f(n)	f(n+1)/f(n)	f(n+2)/f(n)	f(n+3)/f(n)
0	1
1	1	1
2	2	2	2
3	3	1.5	3	3
4	5	1.666666667	2.5	5
5	8	1.6	2.666666667	4
6	13	1.625	2.6	4.333333333
7	21	1.615384615	2.625	4.2
8	34	1.619047619	2.615384615	4.25
9	55	1.617647059	2.619047619	4.230769231
10	89	1.618181818	2.617647059	4.238095238
11	144	1.617977528	2.618181818	4.235294118
12	233	1.618055556	2.617977528	4.236363636
13	377	1.618025751	2.618055556	4.235955056
14	610	1.618037135	2.618025751	4.236111111
15	987	1.618032787	2.618037135	4.236051502
16	1597	1.618034448	2.618032787	4.236074271
17	2584	1.618033813	2.618034448	4.236065574
18	4181	1.618034056	2.618033813	4.236068896
19	6765	1.618033963	2.618034056	4.236067627
20	10946	1.618033999	2.618033963	4.236068111
21	17711	1.618033985	2.618033999	4.236067926
22	28657	1.61803399	2.618033985	4.236067997
23	46368	1.618033988	2.61803399	4.23606797
24	75025	1.618033989	2.618033988	4.23606798
25	121393	1.618033989	2.618033989	4.236067976
26	196418	1.618033989	2.618033989	4.236067978
27	317811	1.618033989	2.618033989	4.236067977
28	514229	1.618033989	2.618033989	4.236067978
29	832040	1.618033989	2.618033989	4.236067977
30	1346269	1.618033989	2.618033989	4.236067978

Table 1: The Fibonacci sequence and ratios of its terms

Open the XLS file to explore this problem. In Table 1, we have the first thirty Fibonacci terms and the ratios of sequential terms, every second term and every third term. The values to which these ratios tend to converge are explored.

Careful observation leads to the hypothesis that the ratio of sequential Fibonacci terms approaches the value known as the Golden Ratio and given by:

Experimentation in the spreadsheet file also leads to the realization that:

Which leads us to hypothesize that the ratio of every second term approaches the value:

Again, experimentation within the spreadsheet leads to another realization that:

Thus, we have the ratio of every third term approaching this value, which should be given by:

Setting and , we have the Lucas Sequence (Table 2).

n	f(n)	f(n+1)/f(n)	f(n+2)/f(n)	f(n+3)/f(n)
0	1
1	3	3
2	4	1.333333	4
3	7	1.75	2.333333	7
4	11	1.571429	2.75	3.666667
5	18	1.636364	2.571429	4.5
6	29	1.611111	2.636364	4.142857
7	47	1.62069	2.611111	4.272727
8	76	1.617021	2.62069	4.222222
9	123	1.618421	2.617021	4.241379
10	199	1.617886	2.618421	4.234043
11	322	1.61809	2.617886	4.236842
12	521	1.618012	2.61809	4.235772
13	843	1.618042	2.618012	4.236181
14	1364	1.618031	2.618042	4.236025
15	2207	1.618035	2.618031	4.236084
16	3571	1.618034	2.618035	4.236062
17	5778	1.618034	2.618034	4.23607
18	9349	1.618034	2.618034	4.236067
19	15127	1.618034	2.618034	4.236068
20	24476	1.618034	2.618034	4.236068
21	39603	1.618034	2.618034	4.236068
22	64079	1.618034	2.618034	4.236068
23	103682	1.618034	2.618034	4.236068
24	167761	1.618034	2.618034	4.236068
25	271443	1.618034	2.618034	4.236068
26	439204	1.618034	2.618034	4.236068
27	710647	1.618034	2.618034	4.236068
28	1149851	1.618034	2.618034	4.236068
29	1860498	1.618034	2.618034	4.236068
30	3010349	1.618034	2.618034	4.236068

Table 2: The Lucas sequence and ratios of its terms

The Lucas sequence can be explored by clicking on the Lucas tab in the Excel spreadsheet. Although the initial values are different, we see that the ratios tend to converge to the same values as those of the Fibonacci Sequence.

Theorem: The ratios of successive terms in a Fibonacci sequence converge to the Golden Ratio. The ratios of pairs of second terms converge to the valueand the ratios of pairs of third terms converge to the value.