Write-up 12

Problem: Generate a Fibonnaci sequence in the first column using f(0)=1,  f(1)=1, f(n) = f(n-1) + f(n-2)

Let's look at the Fibonnaci sequence and the ratio of each pair of adjacent terms

Seq.       Fibonnaci        The ratio           Second             Third

Let's look at the graph of a Fibonnaci sequence and the ratios of adjacent terms, every second term, and every third term.

The ratio of each pair of adjacent terms in the Fibonnaci sequence is 1.618033989 and the ratio of every second term is 2.618033989

and the ratio of every third term is 4.236067977.

Lucas Sequence

Let's look at  a Lucas Sequence f(0)=1, f(1)=3, f(n) = f(n-1) + f(n-2).

Seq.      Lucas        Ratio    Second        Third

Let's look at the graph of a Lucas sequence and the ratios of adjacent terms, every second term, and every third term.

Although the ratio of each pair of adjacent terms in the Lucas sequence is different at several terms with the Fibonnaci sequence, it's the same in the long run.

Similarly, the ratio of every second and third term is the same in the long run between the Lucas sequence and the Fibonnaci sequence.