A sequence of rational numbers is described as follows:
$$\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots$$
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let $x_{n}$ and $y_{n}$ be, respectively, the numerator and the denominator of the $m$ th fraction $r_{n}=x_{n} / y_{n}$
\begin{equation}\begin{array}{l}{\text { a. Verify that } x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1 \text { and, more }} \\ {\text { generally, that if } a^{2}-2 b^{2}=-1 \text { or }+1, \text { then }}\end{array}\end{equation}
$$(a+2 b)^{2}-2(a+b)^{2}=+1 \quad \text { or } \quad-1$$
respectively.
\begin{equation}\begin{array}{l}{\text { b. The fractions } r_{n}=x_{n} / y_{n} \text { approach a limit as } n \text { increases. }} \\ {\text { What is that limit? (Hint: Use part (a) to show that }} \\ {r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2} \text { and that } y_{n} \text { is not less than } n \text { . }}\end{array}\end{equation}