Ronnachai Panapoi

In this write-up, I have an opportunity to see the useful of spreadsheet in Mathematics Exploration. I explore some properties of Fibonacci and Lucas sequence.

First of all, I will give the definition of the Fibonacci sequence.

We define a sequence of numbers as follows:

f₁ = 1

f₂ = 1

f₃ = 1 + 1 = 2

f₄ = 1 + 2 = 3

f₅ = 2 + 3 = 5

and, in general, if n 3, then

f_n = f_n-1 + f_n-2

The number f_n is called the nth Fibonacci number.

The table below shows the first 30 terms of Fibonacci sequence.

n	f(n)
1	1
2	1
3	2
4	3
5	5
6	8
7	13
8	21
9	34
10	55
11	89
12	144
13	233
14	377
15	610
16	987
17	1597
18	2584
19	4181
20	6765
21	10946
22	17711
23	28657
24	46368
25	75025
26	121393
27	196418
28	317811
29	514229
30	832040

We will construct this sequence in spreadsheet. Particularly in this investigation, we will use EXCEL as a tool.

n	f(n)	f(n-1)	f(n-2)
1	1
2	1	1
3	2	1	1
4	3	2	1
5	5	3	2
6	8	5	3
7	13	8	5
8	21	13	8
9	34	21	13
10	55	34	21
11	89	55	34
12	144	89	55
13	233	144	89
14	377	233	144
15	610	377	233
16	987	610	377
17	1597	987	610
18	2584	1597	987
19	4181	2584	1597
20	6765	4181	2584
21	10946	6765	4181
22	17711	10946	6765
23	28657	17711	10946
24	46368	28657	17711
25	75025	46368	28657
26	121393	75025	46368
27	196418	121393	75025
28	317811	196418	121393
29	514229	317811	196418
30	832040	514229	317811

Click HERE to create the sequence in excel file.

Now, we will investigate some relation between the terms of the sequence.

To begin with, we will first explore some relation of the ratio of each pair of adjacent terms in this sequence.

n	f(n)	f(n-1)	f(n)/f(n-1)
1	1
2	1	1	1
3	2	1	2
4	3	2	1.5
5	5	3	1.666667
6	8	5	1.6
7	13	8	1.625
8	21	13	1.615385
9	34	21	1.619048
10	55	34	1.617647
11	89	55	1.618182
12	144	89	1.617978
13	233	144	1.618056
14	377	233	1.618026
15	610	377	1.618037
16	987	610	1.618033
17	1597	987	1.618034
18	2584	1597	1.618034
19	4181	2584	1.618034
20	6765	4181	1.618034
21	10946	6765	1.618034
22	17711	10946	1.618034
23	28657	17711	1.618034
24	46368	28657	1.618034
25	75025	46368	1.618034
26	121393	75025	1.618034
27	196418	121393	1.618034
28	317811	196418	1.618034
29	514229	317811	1.618034
30	832040	514229	1.618034

Click HERE to explore more terms if you want.

Now, we will show that the ratios of the adjacent terms in Fibonacci sequence equal to a specific value. Our goal is to find what such specific value is. Suppose for , , for some real number A.

Since, ,

and .

Therefore, .

We then take the limit of both sides as .

We get

We will investigate the ratios of every second terms in the sense that if they seem like the previous value.

n	f(n)	f(n-2)	f(n)/f(n-2)
1	1
2	1
3	2	1	2
4	3	1	3
5	5	2	2.5
6	8	3	2.66666667
7	13	5	2.6
8	21	8	2.625
9	34	13	2.61538462
10	55	21	2.61904762
11	89	34	2.61764706
12	144	55	2.61818182
13	233	89	2.61797753
14	377	144	2.61805556
15	610	233	2.61802575
16	987	377	2.61803714
17	1597	610	2.61803279
18	2584	987	2.61803445
19	4181	1597	2.61803381
20	6765	2584	2.61803406
21	10946	4181	2.61803396
22	17711	6765	2.618034
23	28657	10946	2.61803399
24	46368	17711	2.61803399
25	75025	28657	2.61803399
26	121393	46368	2.61803399
27	196418	75025	2.61803399
28	317811	121393	2.61803399
29	514229	196418	2.61803399
30	832040	317811	2.61803399

Click HERE to see and explore more in EXCEL file.

From the above table, we can see that the ratios of every second terms are equal to some specific value.

Also, we can plot graph to see the trend as the following.

Likewise, the previous finding of ratios of adjacent terms, we suppose for , let , for some real number A.

Since, ,

Therefore, .

We then take the limit of both sides as .

Similarly, we have

Next, we will explore another sequence as a result from changing the value of the second term in Fibonacci sequence as 3. We call the new sequence that the Lucas sequence.

This table below shows the first 30 terms of Lucas sequence.

n	L(n)
1	1
2	3
3	4
4	7
5	11
6	18
7	29
8	47
9	76
10	123
11	199
12	322
13	521
14	843
15	1364
16	2207
17	3571
18	5778
19	9349
20	15127
21	24476
22	39603
23	64079
24	103682
25	167761
26	271443
27	439204
28	710647
29	1149851
30	1860498

Prior to investigating the ratios of successive terms in Lucas sequence, we will find some relations between Fibonacci and Lucas sequences.

n	1	2	3	4	5	6	…
	1	1	2	3	5	8	…
	1	3	4	7	11	18	…

From the table, we can observe that

We can prove this fact by using Mathematical Induction.

Now, we will explore the ratios of Lucas sequence.

This following table displays the ratios of successive terms in Lucas sequence. It is easy for us to observe some pattern.

n	L(n)	L(n+1)/L(n)
1	1	3
2	3	1.33333333
3	4	1.75
4	7	1.57142857
5	11	1.63636364
6	18	1.61111111
7	29	1.62068966
8	47	1.61702128
9	76	1.61842105
10	123	1.61788618
11	199	1.61809045
12	322	1.61801242
13	521	1.61804223
14	843	1.61803084
15	1364	1.61803519
16	2207	1.61803353
17	3571	1.61803416
18	5778	1.61803392
19	9349	1.61803401
20	15127	1.61803398
21	24476	1.61803399
22	39603	1.61803399
23	64079	1.61803399
24	103682	1.61803399
25	167761	1.61803399
26	271443	1.61803399
27	439204	1.61803399
28	710647	1.61803399
29	1149851	1.61803399
30	1860498	1.61803399

Click HERE to see these data in EXCEL file.

We will show this fact with similar way as we have done with Fibonacci sequence. Anyhow, I will use the relation of .

Previously, we know that which corresponds with our observation.

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