I'd really love your help with finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents.
With the recursion for continued fraction $( p_0=a_0, q_0=1, p_{-1}=1, q_{-1}=o), q_s=a_sq_{s-1}+q_{s-2},p_s=a_sp_{s-1}+p_{s-2}$. I found out the $p_k=1,2,5,7,12,32, q_k=1,1,3,4,7,18 $ (anything special about these series? perhaps I did a mistake?), and I know that $r=\lim_{c_k}=\lim\frac{p_k}{q_k}$, but I can't see anything special about $p_k, q_k$ or the realtion between them, Any help?
Thanks a lot!