6 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 25 terms of the series is 3500 .
- Show that \(a + 12 d = 140\).
- The fifth term of this series is 100 .
Find the value of \(d\) and the value of \(a\).
- The \(n\)th term of this series is \(u _ { n }\). Given that
$$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$
find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
(3 marks)