Proving rn = 2 - 1/(rn-2 + 1): Fibonacci Sequence Homework

[hy23]

Homework Statement

So the Fibonacci Sequence is defined by

a_n = a_n-1 + a_n-2
a₁=1, a₂=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...

 

[LeonhardEuler]

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

 

[hy23]

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]

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)?

 

[hy23]

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]

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

 

[hy23]

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, let's 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]

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 r_n in terms of r_n-1. Do you see how to make a substitution to express r_n-1 in terms of r_n-2? Also, learn how to type equations. You can click on them to see how they are typed.

 

[hy23]

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

thanks for your time though