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.