## Limit of Ratio of Successive Terms for Generalized Fibonacci Sequence

Th Fibonacci sequence is generated by the rule
$a_{n+1} = a_n + a_{n-1}$
with
$a_0 =a_1 =1$
. In English we add two terms to get the next term.
The ratio of successive Fibonacci numbers tends to a limit
$\frac{a_{n+1}}{a+n} = \frac{ \sqrt{5} -1}{2}$
as
$n \rightarrow \infty$
Consider the 'generalized Fibonacci sequence' defined by the rule
$a_{n+1} =A a_n +B a_{n-1} , \: A, B > 0$

Does the ration of successive terms to to a limit for this sequence? Yes it does. Divide the rule that generates this sequence by
$a_n$
to give
$\frac{a_{n+1}}{a_n} =A +B \frac{a_{n-1}}{a_n}$

As
$n \rightarrow \infty$
, if
$\frac{a_{n+1}}{a_n} \rightarrow l$
then
$\frac{a_n}{a_{n-1}} \rightarrow \frac{1}{l}$

Then
$\frac{a_{n+1}}{a_n} =A +B \frac{a_{n-1}}{a_n}$
becomes
$l =A +\frac{B}{l}$

Multiply by
$l$
and rearrange to obtain
$l^2 - Al -B=0$

The solutions are
$l= \frac{A \pm \sqrt{A^2 +4B}}{2}$

The positive option gives the correct value of
$l$
. The negative option gives a negative vale. This is wrong since
$A,B >0$
so
$A < \sqrt{A^2 +4B}$
.