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).

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.

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

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.

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.

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.

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

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