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Fibonacci sequences

Joshua DuMont

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The Fibonacci sequence is defined by f_n = f_n-1+ f_n-2, f₀ = 1, f₁= 1. We could just as easily talk about the sequence where we add the two previous terms to get the next term and start with different initial conditions. A table giving the first 30 terms of three of these sequences is shown along with the ratios of successive terms.

n	fn		gn		hn
0	0		4		5
1	1		7		34
2	1	1	11	1.57142857	39	1.14705882
3	2	2	18	1.63636364	73	1.87179487
4	3	1.5	29	1.61111111	112	1.53424658
5	5	1.66666667	47	1.62068966	185	1.65178571
6	8	1.6	76	1.61702128	297	1.60540541
7	13	1.625	123	1.61842105	482	1.62289562
8	21	1.61538462	199	1.61788618	779	1.61618257
9	34	1.61904762	322	1.61809045	1261	1.61874198
10	55	1.61764706	521	1.61801242	2040	1.61776368
11	89	1.61818182	843	1.61804223	3301	1.61813725
12	144	1.61797753	1364	1.61803084	5341	1.61799455
13	233	1.61805556	2207	1.61803519	8642	1.61804905
14	377	1.61802575	3571	1.61803353	13983	1.61802823
15	610	1.61803714	5778	1.61803416	22625	1.61803619
16	987	1.61803279	9349	1.61803392	36608	1.61803315
17	1597	1.61803445	15127	1.61803401	59233	1.61803431
18	2584	1.61803381	24476	1.61803398	95841	1.61803387
19	4181	1.61803406	39603	1.61803399	155074	1.61803404
20	6765	1.61803396	64079	1.61803399	250915	1.61803397
21	10946	1.618034	103682	1.61803399	405989	1.618034
22	17711	1.61803399	167761	1.61803399	656904	1.61803399
23	28657	1.61803399	271443	1.61803399	1062893	1.61803399
24	46368	1.61803399	439204	1.61803399	1719797	1.61803399
25	75025	1.61803399	710647	1.61803399	2782690	1.61803399
26	121393	1.61803399	1149851	1.61803399	4502487	1.61803399
27	196418	1.61803399	1860498	1.61803399	7285177	1.61803399
28	317811	1.61803399	3010349	1.61803399	11787664	1.61803399
29	514229	1.61803399	4870847	1.61803399	19072841	1.61803399

Notice that the ratio of a term to the previous term seems to approach a particular value, and that that value is the same for all three sets of initial conditions.

Define =Φ

This means = Φ since we have our recursive formula for f_n.

We can rearrange and say that as :

f_n-1+f_n-2= Φ *f_n-1

f_n-2= Φ *f_n-1- f_n-1

f_n-2=( Φ - 1)*f_n-1

=f_n-1

=

As , =.

We can conclude that:

Φ =

Φ *( Φ - 1)=1

Φ ²- Φ -1=0

Φ = ≈1.618

{Note Φ = is unreasonable as the terms in sequence will be positive}

This process did not involve the initial conditions f₀ and f₁, so it will be the same for different choices for the first two terms.

We could also consider ratios of every second term or third term etc.

n	fn
0	0
1	1
2	1	1
3	2	2	2
4	3	1.5	3	3
5	5	1.66666667	2.5	5	5
6	8	1.6	2.66666667	4	8
7	13	1.625	2.6	4.33333333	6.5
8	21	1.61538462	2.625	4.2	7
9	34	1.61904762	2.61538462	4.25	6.8
10	55	1.61764706	2.61904762	4.23076923	6.875
11	89	1.61818182	2.61764706	4.23809524	6.84615385
12	144	1.61797753	2.61818182	4.23529412	6.85714286
13	233	1.61805556	2.61797753	4.23636364	6.85294118
14	377	1.61802575	2.61805556	4.23595506	6.85454545
15	610	1.61803714	2.61802575	4.23611111	6.85393258
16	987	1.61803279	2.61803714	4.2360515	6.85416667
17	1597	1.61803445	2.61803279	4.23607427	6.85407725
18	2584	1.61803381	2.61803445	4.23606557	6.85411141
19	4181	1.61803406	2.61803381	4.2360689	6.85409836
20	6765	1.61803396	2.61803406	4.23606763	6.85410334
21	10946	1.618034	2.61803396	4.23606811	6.85410144
22	17711	1.61803399	2.618034	4.23606793	6.85410217
23	28657	1.61803399	2.61803399	4.236068	6.85410189
24	46368	1.61803399	2.61803399	4.23606797	6.854102
25	75025	1.61803399	2.61803399	4.23606798	6.85410196
26	121393	1.61803399	2.61803399	4.23606798	6.85410197
27	196418	1.61803399	2.61803399	4.23606798	6.85410196
28	317811	1.61803399	2.61803399	4.23606798	6.85410197
29	514229	1.61803399	2.61803399	4.23606798	6.85410197

These also seem to have limits. More than that it seems that the limits have the same pattern as the Fibonacci sequence itself. We guess that:

=+

By our recursive formula for f_n, =

Since the values n-1 and n-k, and the values n and n-(k-1) are the same distance apart we can say that in the limits:

=.

= for similar reasons.

So, =+ as we guessed.