So the Fibonacci Sequence is defined by
an = an-1 + an-2
We are more interested in the sequence of ratios of subsequent terms of the Fibonacci sequence
define rn = an+1 / an
How do we prove that..
rn = 2 - 1/(rn-2 + 1)
for all n>2
The Attempt at a Solution
I attempted to manipulate rn = an+1 / an
by substituting an-1 + an-2 for an
and doing similar substitutions for other terms as well, I think that by making correct substitutions, it will eventually simplify to what we want to prove. So far, no luck...