6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).
- Given that \(u _ { k } = 140\), find the value of \(k\).
A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
- Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
- Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).