Condition for following sequence (given linear recurrence) to be bounded below

Question

Given the sequence:

$x_{n+1} + x_{n} - 6x_{n-1} = 0$

I need to give a necessary and sufficient condition on $x_{0}$ and $x_{1}$ such that the sequence is bounded below.

I am confused about "necessary and sufficient". Is it enough to say that for the sequence to be bounded below by the first term it has to be increasing, so $x_{1} > x_{0}$?

Answer 1

Such recurrence relations can be solved by solving the characteristic quadratic equation, in this case

$$x^2+x-6=0$$

The roots are $-3$ and $2$ and the general solution is $$x_n=c_1\cdot (-3)^n+c_2\cdot 2^n$$

This is bounded from below if and only if $c_1=0$ and $c_2\ge 0$, so we have the general solution $$x_n=c_2\cdot 2^n$$

Inserting $n=0$ and $n=1$ , we have $x_0=c_2$ and $x_1=2c_2$

Hence the condition is $x_1=2x_0\ge 0$