7 The members of the family of the sequences \(\left\{ u _ { n } \right\}\) satisfy the recurrence relation $$u _ { n + 1 } = 10 u _ { n } - u _ { n - 1 } \text { for } n \geqslant 1$$

1. Determine the general solution of (*).
2. The sequences \(\left\{ a _ { n } \right\}\) and \(\left\{ b _ { n } \right\}\) are members of this family of sequences, corresponding to the initial terms \(a _ { 0 } = 1 , a _ { 1 } = 5\) and \(b _ { 0 } = 0 , b _ { 1 } = 2\) respectively.
   (a) Find the next two terms of each sequence.
   (b) Prove that, for all non-negative integers \(n , \left( a _ { n } \right) ^ { 2 } - 6 \left( b _ { n } \right) ^ { 2 } = 1\).
   (c) Determine \(\lim _ { n \rightarrow \infty } \left( \frac { a _ { n } } { b _ { n } } \right)\). \section*{OCR} Oxford Cambridge and RSA