Investigating f(n) = f(n-1) + f(n-2) for values that will generate the Lucas Sequence

by

Kimberly Burrell

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 with the Golden Ratio (1.6180339) regardless of the values of f(0) and f(1).

 1 3 3 1.33333 4 1.75 7 1.57143 11 1.63636 18 1.61111 29 1.62069 47 1.61702 76 1.61842 123 1.61789 199 1.61809 322 1.61801 521 1.61804 843 1.61803 1364 1.61804 2207 1.61803 3571 1.61803 5778 1.61803 9349 1.61803 15127 1.61803 24476 1.61803 39603 1.61803 64079 1.61803 103682 1.61803 167761 1.61803 271443 1.61803

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

 1 4 3 2.33333 4 2.75 7 2.57143 11 2.63636 18 2.61111 29 2.62069 47 2.61702 76 2.61842 123 2.61789 199 2.61809 322 2.61801 521 2.61804 843 2.61803 1364 2.61804 2207 2.61803 3571 2.61803 5778 2.61803 9349 2.61803 15127 2.61803 24476 2.61803 39603 2.61803 64079 2.61803 103682 2.61803 167761 2.61803 271443 2.61803

We will now investigate the limit of the ratio of every third term of the given sequence.

 1 7 3 3.66667 4 4.5 7 4.14286 11 4.27273 18 4.22222 29 4.24138 47 4.23404 76 4.23684 123 4.23577 199 4.23618 322 4.23602 521 4.23608 843 4.23606 1364 4.23607 2207 4.23607 3571 4.23607 5778 4.23607 9349 4.23607 15127 4.23607 24476 4.23607 39603 4.23607 64079 4.23607 103682 4.23607 167761 4.23607 271443 4.23607

Once again one can observe that we find the limit is the same as the limit 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) = 5 and f(1) = 7.

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

 5 1.4 7 1.71429 12 1.58333 19 1.63158 31 1.6129 50 1.62 81 1.61728 131 1.61832 212 1.61792 343 1.61808 555 1.61802 898 1.61804 1453 1.61803 2351 1.61803 3804 1.61803 6155 1.61803 9959 1.61803 16114 1.61803 26073 1.61803 42187 1.61803 68260 1.61803 110447 1.61803 178707 1.61803 289154 1.61803 467861 1.61803 757015 1.61803

Again, one can observe we have found the limit to be 1.6180339887. This sequence does not prove that this will be the value of this limit for all values of f(0) and f(1), but it does indicate 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) = 5 and f(1) = 7.

 5 2.4 7 2.71429 12 2.58333 19 2.63158 31 2.6129 50 2.62 81 2.61728 131 2.61832 212 2.61792 343 2.61808 555 2.61802 898 2.61804 1453 2.61803 2351 2.61803 3804 2.61803 6155 2.61803 9959 2.61803 16114 2.61803 26073 2.61803 42187 2.61803 68260 2.61803 110447 2.61803 178707 2.61803 289154 2.61803 467861 2.61803 757015 2.61803

2.6180339887 AGAIN! We are building momentum in our investigation even though we have not proven our conjecture that this limit will always be 2.6180339887.

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

 5 3.8 7 4.42857 12 4.16667 19 4.26316 31 4.22581 50 4.24 81 4.23457 131 4.23664 212 4.23585 343 4.23615 555 4.23604 898 4.23608 1453 4.23606 2351 4.23607 3804 4.23607 6155 4.23607 9959 4.23607 16114 4.23607 26073 4.23607 42187 4.23607 68260 4.23607 110447 4.23607 178707 4.23607 289154 4.23607 467861 4.23607 757015 4.23607

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