Convergence of recursively defined sequence?

Question

I was thinking about this recurrence defined sequence-

$a_{n} = \frac{a_{n-1}+a_{n-2}}{2}$ for $n \geq 3$

Does $(a_{n})$ necessarily converge? It is just mentioned that $a_{0},a_{1}$ are real numbers

I was thinking how to do this?

Any idea?

Answer 1

We have $$2a_n-a_{n-1}-a_{n-2}=0$$

Using this, $$a_n=A\left(-\dfrac12\right)^n+B\cdot1^n$$ where $A,B$ are arbitrary constants

Now $\displaystyle\lim_{n\to\infty}\left(-\dfrac12\right)^n=?$