6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by
$$\begin{gathered}
x _ { 1 } = 1
x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1
\end{gathered}$$
where \(k\) is a constant, \(k \neq 0\)
- Find an expression for \(x _ { 2 }\) in terms of \(k\).
- Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\)
Given also that \(x _ { 3 } = 1\),
- calculate the value of \(k\).
- Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)