Investigating f(n) = f(n-1) + f(n-2) for various values of f(0) and f(1)

by Amy Benson


f(0) = 1 and f(1) = 3

Using this sequence, but changing f(0) and f(1) to f(0) = 1 and f(1) = 3 generates a Lucas Sequence.

Notice that the ratio of adjacent terms remains constant (1.6180339) regardless of the values of f(0) and f(1).

1 3
3 1.33333333333333
4 1.75
7 1.57142857142857
11 1.63636363636364
18 1.61111111111111
29 1.62068965517241
47 1.61702127659574
76 1.61842105263158
123 1.61788617886179
199 1.61809045226131
322 1.61801242236025
521 1.61804222648752
843 1.61803084223013
1364 1.61803519061584
2207 1.6180335296783
3571 1.61803416409969
5778 1.61803392177224
9349 1.61803401433308
15127 1.61803397897799
24476 1.61803399248243
39603 1.61803398732419
64079 1.61803398929446
103682 1.61803398854189
167761 1.61803398882935
271443 1.61803398871955
439204 1.61803398876149
710647 1.61803398874547
1149851 1.61803398875159
1860498 1.61803398874925
3010349 1.61803398875014
4870847 1.6180339887498
7881196 1.61803398874993
12752043 1.61803398874988

The same holds true for the limit of the ratio of every second term as seen below. The limit is again 2.6180339 (the same value found previously for the ratio of every second term with f(0) = 1 and f(1) = 1.

1 4
3 2.33333333333333
4 2.75
7 2.57142857142857
11 2.63636363636364
18 2.61111111111111
29 2.62068965517241
47 2.61702127659574
76 2.61842105263158
123 2.61788617886179
199 2.61809045226131
322 2.61801242236025
521 2.61804222648752
843 2.61803084223013
1364 2.61803519061584
2207 2.6180335296783
3571 2.61803416409969
5778 2.61803392177224
9349 2.61803401433308
15127 2.61803397897799
24476 2.61803399248243
39603 2.61803398732419
64079 2.61803398929446
103682 2.61803398854189
167761 2.61803398882935
271443 2.61803398871955
439204 2.61803398876149
710647 2.61803398874547
1149851 2.61803398875159
1860498 2.61803398874925
3010349 2.61803398875014
4870847 2.6180339887498
7881196 2.61803398874993
12752043 2.61803398874988
20633239 2.6180339887499

Now, we investigate the limit of the ratio of every third term of the given sequence.

1 7
3 3.66666666666667
4 4.5
7 4.14285714285714
11 4.27272727272727
18 4.22222222222222
29 4.24137931034483
47 4.23404255319149
76 4.23684210526316
123 4.23577235772358
199 4.23618090452261
322 4.2360248447205
521 4.23608445297505
843 4.23606168446026
1364 4.23607038123167
2207 4.23606705935659
3571 4.23606832819938
5778 4.23606784354448
9349 4.23606802866617
15127 4.23606795795597
24476 4.23606798496486
39603 4.23606797464839
64079 4.23606797858893
103682 4.23606797708378
167761 4.23606797765869
271443 4.23606797743909
439204 4.23606797752297
710647 4.23606797749093

Here again, we find that the limit is the same as the limit found in our previous example (4.236067977).


Another pair of values for f(0) and f(1)

Let's investigate our Fibonnaci Sequence with f(0) = 3 and f(1) = 7.

We begin our investigation by determining the limit of the ratio of adjacent terms.

3 1.66666666666667
5 1.6
8 1.625
13 1.61538461538462
21 1.61904761904762
34 1.61764705882353
55 1.61818181818182
89 1.61797752808989
144 1.61805555555556
233 1.61802575107296
377 1.61803713527851
610 1.61803278688525
987 1.61803444782168
1597 1.61803381340013
2584 1.61803405572755
4181 1.61803396316671
6765 1.6180339985218
10946 1.61803398501736
17711 1.6180339901756
28657 1.61803398820532
46368 1.6180339889579
75025 1.61803398867044
121393 1.61803398878024
196418 1.6180339887383
317811 1.61803398875432
514229 1.6180339887482
832040 1.61803398875054
1346269 1.61803398874965
2178309 1.61803398874999
3524578 1.61803398874986
5702887 1.61803398874991
9227465 1.61803398874989
14930352 1.6180339887499
24157817 1.61803398874989
39088169 1.6180339887499
63245986 1.61803398874989
102334155 1.61803398874989
165580141 1.61803398874989

Again, we have found the limit to be 1.6180339887. This does not prove that this will be the value of this limit for all values of f(0) and f(1), but it indicates that we are on the right track.

Now, we determine the limit of the ratio of every second term of this sequence with f(0) = 3 and f(1) = 7

3 2.66666666666667
5 2.6
8 2.625
13 2.61538461538462
21 2.61904761904762
34 2.61764705882353
55 2.61818181818182
89 2.61797752808989
144 2.61805555555556
233 2.61802575107296
377 2.61803713527851
610 2.61803278688525
987 2.61803444782168
1597 2.61803381340013
2584 2.61803405572755
4181 2.61803396316671
6765 2.6180339985218
10946 2.61803398501736
17711 2.6180339901756
28657 2.61803398820533
46368 2.6180339889579
75025 2.61803398867044
121393 2.61803398878024
196418 2.6180339887383
317811 2.61803398875432
514229 2.6180339887482
832040 2.61803398875054
1346269 2.61803398874965
2178309 2.61803398874999
3524578 2.61803398874986
5702887 2.61803398874991
9227465 2.61803398874989
14930352 2.6180339887499
24157817 2.61803398874989
39088169 2.6180339887499
63245986 2.61803398874989
102334155 2.61803398874989

2.6180339887 AGAIN. We are building momentum in our investigation even though we haven't proven our conjecture that this limit will always be 2.6180339887.

Lastly, we find the limit of the ratio of every third term of our sequence.

3 4.33333333333333
5 4.2
8 4.25
13 4.23076923076923
21 4.23809523809524
34 4.23529411764706
55 4.23636363636364
89 4.23595505617978
144 4.23611111111111
233 4.23605150214592
377 4.23607427055703
610 4.23606557377049
987 4.23606889564336
1597 4.23606762680025
2584 4.23606811145511
4181 4.23606792633341
6765 4.23606799704361
10946 4.23606797003472
17711 4.23606798035119
28657 4.23606797641065
46368 4.2360679779158
75025 4.23606797734089
121393 4.23606797756049
196418 4.23606797747661
317811 4.23606797750864
514229 4.23606797749641
832040 4.23606797750108
1346269 4.2360679774993
2178309 4.23606797749998
3524578 4.23606797749972
5702887 4.23606797749982
9227465 4.23606797749978
14930352 4.23606797749979
24157817 4.23606797749979
39088169 4.23606797749979
63245986 4.23606797749979

Our limit retains the same value as in our two previous examples. We can conjecture that the value will always be 4.236067977.


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