Let's generate a Fibonnaci sequence in the first column using f(0)=1, f(1)=1, f(n)=f(n-1)+f(n-2)

Construct the ratio of each pair of adjacent term in the Fibonnaci sequence.


Let's investigate what happens when we get the ratio of every second, third and fourth term.

If you look at the table, we can find a pattern among the convergence of ratios. There also is a sequence like Fibonnaci sequence.

the convergence of Ratio(n) = the convergence of Ratio(n-1) +the convergence of Ratio(n-2)


Now, explore the case where f(0)=1 and f(1)=3, then this sequence is a Lucas Sequence.

This sequence also have the same limit of the ratio of successive terms.

Therefore, no matter what you choose number for f(0) and f(1) with Fibonnaci sequence formula, it will converge to the golden ratio.