I was given this question. It asked to state which of the sequences, $a_n$, $b_n$ and $u_n$ are divergent, convergent or periodic.
The given info is:
$$u_1 = 1 \\ u_n\cdot u_{n+1} =2\\ a_n = u_{n+1} + u_n\\ b_n = u_{n+1} - u_n$$
I got $u_n$ to be $1,2,1,2,\dots$
$a_n$ to be $3,3,3,3,3,\dots$
$b_n$ to be $1,-1,1,-1,1,\dots$
So granted that I worked these out correctly. I'm guessing $u_n$ and $b_n$ are periodic, however I'm not sure about $a_n$.