Problems

The sequence \(a_n\) is defined by \(a_0 = 1\) , \(a_1 = 1\) , and $$ a_n = {1 + a_{n - 1}^2 \over a_{n - 2} } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( n \ge 2 ) . $$ Prove by induction that $$ a_n = 3 a_{n - 1} - a_{n - 2} \ \ \ \ \ \ \ \ \ \ \ ( n \ge2 ) . $$ Hence show that $$ a_n = {\alpha^{2 n - 1} + \alpha^{- ( 2 n - 1 )} \over \sqrt 5} \ \ \ \ \ \ (n\ge1), $$ where \(\displaystyle{\alpha = {1 + \sqrt 5 \over 2}}\).