I am setting out to find a recursive formula for the ratio of 2 successive terms of the fibonacci sequence.
The fibonacci sequence recursive formula is f(n)=f(n-1) + f(n-2), f(1)=1, f(1)=1
Some thoughts on the fibonacci numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
1/1=1
2/1=2
3/2=1.5
5/3=1.66666
8/5=1.6
13/8=1.625
21/13=1.6153846154
34/21=… and so on
Here is some code I typed up in python that graphs the ratio of two successive terms of the fibonacci sequence.
This is a line graph with all of the points visible.
This is a line graph showing most of the points, but only from y=1.618 to y=1.6181. This pattern seems to be oscillating and converges to the golden ratio.
This formula will be represented by r(n) with r(1)=1, r(2)=2, r(3)=1.5, r(4)=1.666667, r(5)=1.6, r(6)=1.625 and so on.
r(n) = f(n+1)/f(n), so writing this in terms of r(n-1) seems to be a little tricky.
Maybe something like r(n) = r(n-1) + (-1)^n*r(n-1)
After doing some research, there is a recursive sequence for rn that is only in terms of r(n-1). The sequence is rn = 1 + 1/r(n-1). This sequence oscillates and is bounded by 1 ≤ rn ≤ 2. Even when the difference between r(n) and r(n+1) is very small as n goes to infinity, the sequence will still oscillate. I found out while trying to figure out a formula for r(n) that to go from r(n) to r(n+1) is +1/f(n)f(n+1) then -1/f(n+1)f(n+2) then +1/f(n+2)f(n+3) and so on (fn is the fibonacci series). This series does converge to a limit and that limit is the golden ratio or (1+sqrt(5))/2. After a quick search on what a Cauchy sequence is, I believe r(n) is one. From what I observe and read, a Cauchy sequence is a sequence that oscillates and the outputs of the sequence become closer and closer as n becomes larger and larger.