7 Binet's formula for the \(n\)th Fibonacci number is given by \(\mathrm { F } _ { \mathrm { n } } = \frac { 1 } { \sqrt { 5 } } \left( \alpha ^ { \mathrm { n } } - \beta ^ { \mathrm { n } } \right)\) for \(n \geqslant 0\), where \(\alpha\) and \(\beta\) (with \(\alpha > 0 > \beta\) ) are the roots of \(x ^ { 2 } - x - 1 = 0\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Consider the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\), where \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\) for \(n \geqslant 0\).
   1. Determine the values of \(S _ { 2 }\) and \(S _ { 3 }\).
   2. Show that \(S _ { n + 2 } = S _ { n + 1 } + S _ { n }\) for \(n \geqslant 0\).
   3. Deduce that \(S _ { n }\) is an integer for all \(n \geqslant 0\).
3. A student models the terms of the sequence \(\left\{ \mathrm { S } _ { \mathrm { n } } \right\}\) using the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\).
   1. Explain why this formula is unsuitable for every \(n \geqslant 1\).
   2. Considering the cases \(n\) even and \(n\) odd separately, state a modification of the formula \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } }\), other than \(\mathrm { T } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } }\), such that \(\mathrm { T } _ { \mathrm { n } } = \mathrm { S } _ { \mathrm { n } }\) for all \(n \geqslant 1\).