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).

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.

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

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.

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

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.

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

return