(i) Find a pair of positive integers, $x _ { 1 }$ and $y _ { 1 }$, that solve the equation $$\left( x _ { 1 } \right) ^ { 2 } - 2 \left( y _ { 1 } \right) ^ { 2 } = 1$$ (ii) Given integers $a , b$, we define two sequences $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ and $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ by setting $$x _ { n + 1 } = 3 x _ { n } + 4 y _ { n } , \quad y _ { n + 1 } = a x _ { n } + b y _ { n } , \quad \text { for } n \geqslant 1$$ Find positive values for $a , b$ such that $$\left( x _ { n + 1 } \right) ^ { 2 } - 2 \left( y _ { n + 1 } \right) ^ { 2 } = \left( x _ { n } \right) ^ { 2 } - 2 \left( y _ { n } \right) ^ { 2 } .$$ (iii) Find a pair of integers $X , Y$ which satisfy $X ^ { 2 } - 2 Y ^ { 2 } = 1$ such that $X > Y > 50$. (iv) (Using the values of $a$ and $b$ found in part (ii)) what is the approximate value of $x _ { n } / y _ { n }$ as $n$ increases?
(i) [2 marks] A fairly obvious pair $\left( x _ { 1 } , y _ { 1 } \right)$ that satisfy $\left( x _ { 1 } \right) ^ { 2 } - 2 \left( y _ { 1 } \right) ^ { 2 } = 1$ is $x _ { 1 } = 3$ and $y _ { 1 } = 2$. [0pt]
\section*{2. For ALL APPLICANTS.}
(i) Find a pair of positive integers, $x _ { 1 }$ and $y _ { 1 }$, that solve the equation
$$\left( x _ { 1 } \right) ^ { 2 } - 2 \left( y _ { 1 } \right) ^ { 2 } = 1$$
(ii) Given integers $a , b$, we define two sequences $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ and $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ by setting
$$x _ { n + 1 } = 3 x _ { n } + 4 y _ { n } , \quad y _ { n + 1 } = a x _ { n } + b y _ { n } , \quad \text { for } n \geqslant 1$$
Find positive values for $a , b$ such that
$$\left( x _ { n + 1 } \right) ^ { 2 } - 2 \left( y _ { n + 1 } \right) ^ { 2 } = \left( x _ { n } \right) ^ { 2 } - 2 \left( y _ { n } \right) ^ { 2 } .$$
(iii) Find a pair of integers $X , Y$ which satisfy $X ^ { 2 } - 2 Y ^ { 2 } = 1$ such that $X > Y > 50$.\\
(iv) (Using the values of $a$ and $b$ found in part (ii)) what is the approximate value of $x _ { n } / y _ { n }$ as $n$ increases?