TITLE: How can I systematically find a solution to this problem?
QUESTION [5 upvotes]: While doing some self-study, a friend posed this problem to me:

Let $a$ be a sequence, defined as following:
$$a_0 = 0,\quad a_1= 1, \qquad a_{n+2}=\frac{a_n+a_{n+1}}2$$
Figure out, whether $a$ converges, and if yes to which value. Find a closed form for $a_n$. If you want to, you may also have a look at the common case with $a_0= \alpha$ and $a_1 = \beta$.

It's easy to see, that this converges to $\frac2 3$, and the other parts are also easy to solve with quessing a result and afterwards proving it correct, but I dislike this approach, as it is not productive if the solution isn't obvious.
Is there a systematically approach for such problems? I would be really happy if somebody can explain me, how to systematically solve problems like this.

REPLY [2 votes]: I realize you're asking for a systematic answer to the problem, and others have supplied such, but there is a neat correspondence with the binary representation of 1/3. 
Every iteration of the recurrence, you're dividing everything by two each time, you're going up and down each time by a negative power of 2, $1 - 1/2 + 1/4 - 1/8 + ...$ or $1/2 + 1/8 + 1/32 + ...$ essentially creating a binary encoding .1010101... which, as a geometric series comes out to $${1\over 2}\cdot{1\over 1- {1\over 4}} = {2\over 3}.$$