9. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = \frac { k \left( a _ { n } + 2 \right) } { a _ { n } } \quad n \in \mathbb { N }$$
where \(k\) is a constant.
Given that
- the sequence is a periodic sequence of order 3
- \(a _ { 1 } = 2\)
- show that
$$k ^ { 2 } + k - 2 = 0$$
For this sequence explain why \(k \neq 1\)Find the value of
$$\sum _ { r = 1 } ^ { 80 } a _ { r }$$
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