8 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { \lambda }$$ where \(\lambda\) is a constant greater than 1 . Prove by mathematical induction that, for \(n \geqslant 2\), $$a _ { n } \geqslant 2 ^ { \mathrm { g } ( n ) }$$ where \(g ( n ) = \lambda ^ { n - 1 }\). Prove also that, for \(n \geqslant 2 , \frac { a _ { n + 1 } } { a _ { n } } > 2 ^ { ( \lambda - 1 ) \mathrm { g } ( n ) }\).