Find Limit of Fibonacci Sequence | Determine L

[Punkyc7]

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 without 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?

 

[tiny-tim]

Hi Punkyc7 !

If there is a limit, then you can assume that x_n = x_n-1

 

[Punkyc7]

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]

How can you just make a jump and say there is a $\sqrt{5}$ in there

quadratic equation?

 

[Punkyc7]

Ok I am not sure how you got there but this is what I have so farlet $x_{n}$ = $f_{n+1}$/$f_{n}$ 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]

put x_n = x_n-1

 

[Punkyc7]

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 doesn't look very pretty to solve...

 

[tiny-tim]

erm …

you'll also need f_n+1 = … ? ​

 

[Punkyc7]

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]

This is the Fibonacci sequence!

So f_n+1 = ? ​

 

[Punkyc7]

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]

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:​

 

[Punkyc7]

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=0How 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?

 

[eumyang]

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.