3 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and
$$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { 3 }$$
- Prove by mathematical induction that \(\ln a _ { n } \geqslant 3 ^ { n - 1 } \ln 2\) for all integers \(n \geqslant 2\).
[0pt]
[You may use the fact that \(\ln \left( x + \frac { 1 } { x } \right) > \ln x\) for \(x > 0\).] - Show that \(\ln \mathrm { a } _ { \mathrm { n } + 1 } - \ln \mathrm { a } _ { \mathrm { n } } > 3 ^ { \mathrm { n } - 1 } \ln 4\) for \(n \geqslant 2\).