Find the value of the periodic continued fraction with the terms $1, 3, 4, 3, 2, 3, 4, 3, 2, 3, 4, 3, 2, . . .$
We see that it starts to be periodic after $1$ = $q_0$, i.e, $3,4,3,2$ then $3,4,3,2$ etc...
I know that $x= \frac{A_{k+1}}{B_{k+1}}$ = $\frac{A_{k-1}+y.A_K}{B_{k-1}+y.B_K}$ where $q_{k+1}=y$
but then how do I find my number $x$ = continued fraction?
You have $$X={3+\cfrac{1}{4+\cfrac{1}{3+\cfrac{1}{2+\cfrac{1}{X}}}}}$$ for the periodic part. If you use your formula to compute the right-hand side, you will end up with a quadratic in $X$. Solve for $X$ and note that $X$ must be positive. Don't forget to add in the $1$ at the end.