## 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 \[A < \sqrt{A^2 +4B}\]

.