11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).
- For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
- For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\). - Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
- When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).