# Fibonacci Sequence

Let f$_{n}$ be the Fibonacci sequence and let $x_{n}$ = $f_{n+1}$/$f_{n}$. Given that lim$(x_{n}$)=L exist determine L.

Ok so I know that the limit is $\frac{1+\sqrt{5}}{2}$ from previous experience with the sequence, but I am not sure how do you show that with out writing out a lot of terms and then noticing what I all ready know it is. How do you find the limit of a sequence to a number if your not given any numbers to work with?

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tiny-tim
Homework Helper
Hi Punkyc7 ! If there is a limit, then you can assume that xn = xn-1 true, but how do you ever get a number when you are dealing with f$_{n}$ and f$_{n+1}$. How can you just make a jump and say there is a $\sqrt{5}$ in there

tiny-tim
Homework Helper
How can you just make a jump and say there is a $\sqrt{5}$ in there
quadratic equation? Ok im not sure how you got there but this is what I have so far

let $x_{n}$ = $f_{n+1}$/$f_{n}$ and let lim$(x_{n}$)=L. From here is where I get stuck. I know that every sub sequence of a convergent sequence converges to the same number by some theorem, but I am not sure how that is at all helpful.

Would you do something like
$x_{n}$ = $f_{n+1}$/$f_{n}$
$x_{n-1}$ = $f_{n}$/$f_{n-1}$

tiny-tim
Homework Helper
put xn = xn-1

Ok so you get

= $f_{n}$/$f_{n-1}$ = $f_{n+1}$/$f_{n}$

I think I see where you got the quadratic equation now

$f_{n}$ ^2= $f_{n-1}$ $f_{n+1}$=$f_{n}$ ^2 - $f_{n-1}($ $f_{n+1}$)

to use the quadratic equation is this $f_{n-1}($ $f_{n+1}$) considered b or c?

and that doesnt look very pretty to solve...

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tiny-tim
Homework Helper
erm you'll also need fn+1 = … ? Ok

$f_{n+1}$=$\frac{f_{n}}{f_{n-1}}$

$f_{n}$ ^2 - f$_{n-1}$* $\frac{f_{n}}{f_{n-1}}$

$f_{n}$ ^2 -$f_{n}$=0

is that right?

tiny-tim
Homework Helper
This is the Fibonacci sequence!

So fn+1 = ? oh so
f$_{n+1}$= f$_{n}$ +f$_{n-1}$

do I use that for f$_{n+1}$
Also how do you know when to use what?

tiny-tim
Homework Helper
do I use that for f$_{n+1}$
yes … that should give you your quadratic equation Also how do you know when to use what?
You're told it's a Fibonacci sequence, so you obviously have to use that information somewhere! and now i'm off to bed :zzz:

let $x_{n}$ = $f_{n+1}$/$f_{n}$ and let lim$(x_{n}$)=L. Since we know the sequence converges we can say

$x_{n}$ =$x_{n-1}$ Which Implies

= $f_{n}$/$f_{n-1}$ = $f_{n+1}$/$f_{n}$

$f_{n}$ ^2= $f_{n-1}$ $f_{n+1}$=$f_{n}$ ^2 - $f_{n-1}($ $f_{n+1}$)=0

$f_{n}$ ^2 - f$_{n}$f$_{n-1}$-f$_{n-1}$^2=0

How do you hammer this into the quadratic equation I am thinking the a=1 b=not sure c=not sure ? Also how do you get numbers from this when we don't have a single number?

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eumyang
Homework Helper
let $x_{n}$ = $f_{n+1}$/$f_{n}$ and let lim$(x_{n}$)=L. Since we know the sequence converges we can say

$x_{n}$ =$x_{n-1}$ Which Implies

= $f_{n}$/$f_{n-1}$ = $f_{n+1}$/$f_{n}$
...which also equals L:

$\frac{f_n}{f_{n-1}} = \frac{f_{n+1}}{f_n} = L$

Now take this portion:
$\frac{f_{n+1}}{f_n} = L$

Replace the numerator with its equivalent, and then rewrite as a sum of two fractions. A substitution can be made, and you will end up with an expression on the left side with NO f's. Soon you will see a quadratic equation in terms of L. Solve for L.