# Fibonacci Sequence

## Homework Statement

So the Fibonacci Sequence is defined by

an = an-1 + an-2
a1=1, a2=1

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

none

## 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...

LeonhardEuler
Gold Member
What did you get after the substitution? You seem to be on the right track, but to express the result in terms of rn-2, you probably need to either make the substitutions twice, or find a relation that expresses rn in terms of rn-1, then use this relation to express rn-1 in terms of rn-2

yep I think that's the way to do it, I made the substitution twice but I still don't know which substitutions or which algebraic tricks to use to get that +1 term behind the r(n-2). If you can please show me how

LeonhardEuler
Gold Member
There is an algebraic trick that goes like this:
$$\frac{a}{a+b}=\frac{a+b-b}{a+b}=1-\frac{b}{a+b}$$
The way I did it, I had to use this trick, but I'm not going to give away the answer. Why don't you show what you got when you did the double substitution (unless this gets you to figure it out)?

what you did there was the example shown by my prof, which seems straightforward enough, but the other expression is what I can't get to

LeonhardEuler
Gold Member
What step are you stuck at? What is your expression after you do the substitutions?

oh man I really don't want to type it out

basically after my first substitution a(n-1) + a(n-2) for an
I made a similar substitution for a(n+1) = an + a(n-1)
this doesn't really simplify down to the expression I want, so I said ok, lets keep substituting, so I took a(n-1) and substituted that with a(n-2) + a(n-3), but that just adds more terms that I can't cancel out...AAAAARRRRRGG, I'm so stuck

LeonhardEuler
Gold Member
Here's a hint:
$$r_{n} = \frac{a_{n+1}}{a_{n}} = \frac{a_{n} + a_{n-1}}{a_{n}} = 1 + \frac{a_{n-1}}{a_{n}} = 1 + \frac{1}{r_{n-1}}$$
Can you see what to do next? You have rn in terms of rn-1. Do you see how to make a substitution to express rn-1 in terms of rn-2? Also, learn how to type equations. You can click on them to see how they are typed.

hmm I finally figured out how but probably not the way you were suggesting, I just started from the end and worked backwards to get it