6. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned}
a _ { 1 } & = 4
a _ { n + 1 } & = 5 - k a _ { n } , \quad n \geqslant 1
\end{aligned}$$
where \(k\) is a constant.
- Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\).
Find
- \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
- \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)