The sequence is the solution of the difference equation...

(1)

... with some sort of 'initial value'

. We don't have information regarding r and, only for demonstrative purpose, we will suppose that

. The 'fixed point' of the sequence are the solution of the equation

that are...

(2)

With the hypotheses we have made

is a

*repulsive* fixed point [i.e. a point from which the

tend to diverge...] and

an

*attractive* fixed point [i.e. a point to which the

tend to converge...]. The figure is the case

so that is

...

From the figure we can derive the asintotic behavior of the

as function of

. In particular...

a) if

or

the sequence will diverge to

...

b) if

or

the sequence will converge to

c) if

the sequence will converge to

without oscillations...

Kind regards