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## Homework Statement

So the Fibonacci Sequence is defined by

a

_{n}= a

_{n-1}+ a

_{n-2}

a

_{1}=1, a

_{2}=1

We are more interested in the sequence of ratios of subsequent terms of the Fibonacci sequence

define r

_{n}= a

_{n+1}/ a

_{n}

How do we prove that..

r

_{n}= 2 - 1/(r

_{n-2}+ 1)

for all n>2

## Homework Equations

none

## The Attempt at a Solution

I attempted to manipulate r

_{n}= a

_{n+1}/ a

_{n}

by substituting a

_{n-1}+ a

_{n-2}for a

_{n}

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...