Some questons about recurrence sequences (using a problem).

A quick doubt, lets study the recurrence sequence:

${A}_{n+1}=(4{A}_{n}+2)/{A}_{n}+3$

${A}_{0}<-3$

First of all i do:

${A}_{n+1}-{A}_{n}<0$

If this is true i can say that ${A}_{n}$ decrease. This is true for those ${A}_{n}$ values:

$-3>{A}_{n}>-1$

${A}_{n}>2$

And false (so ${A}_{n}$ increase) for those ${A}_{n}$ values:

${A}_{n}<-3$

$-1<{A}_{n}<2$

the case ${A}_{n}<-3$ interests me.

The limit L can be -1 or -2 but i cannot say it exists for sure because ${A}_{n}$ is not limited and monotone for all the ${A}_{n}$ possible values. For example, the sequence can go from ${A}_{n}>2$ then decrease and go in ${A}_{n}<-3$ then again increase and fall in ${A}_{n}>2$ etc...

Another doubt comes from this fact:

It's ok to remove -1 from the possible values of L because in this case the sequence still growing?

Anyways: It happens so many times that i know the sequence increase or decrease in an interval but i don't know if doing it it will fall in another interval where it starts decreasing or increasing and in this scenario i don't know how to demonstrate if it goes on some limit or just starts to "ping-pong" on different intervals.

Or in other words i don't know how many it decrease/increase so i cannot say if it will go out from the interval i'm considering.

A quick doubt, lets study the recurrence sequence:

${A}_{n+1}=(4{A}_{n}+2)/{A}_{n}+3$

${A}_{0}<-3$

First of all i do:

${A}_{n+1}-{A}_{n}<0$

If this is true i can say that ${A}_{n}$ decrease. This is true for those ${A}_{n}$ values:

$-3>{A}_{n}>-1$

${A}_{n}>2$

And false (so ${A}_{n}$ increase) for those ${A}_{n}$ values:

${A}_{n}<-3$

$-1<{A}_{n}<2$

the case ${A}_{n}<-3$ interests me.

The limit L can be -1 or -2 but i cannot say it exists for sure because ${A}_{n}$ is not limited and monotone for all the ${A}_{n}$ possible values. For example, the sequence can go from ${A}_{n}>2$ then decrease and go in ${A}_{n}<-3$ then again increase and fall in ${A}_{n}>2$ etc...

Another doubt comes from this fact:

It's ok to remove -1 from the possible values of L because in this case the sequence still growing?

Anyways: It happens so many times that i know the sequence increase or decrease in an interval but i don't know if doing it it will fall in another interval where it starts decreasing or increasing and in this scenario i don't know how to demonstrate if it goes on some limit or just starts to "ping-pong" on different intervals.

Or in other words i don't know how many it decrease/increase so i cannot say if it will go out from the interval i'm considering.