A geometric progression has first term \(3 a\) and common ratio \(r\). A second geometric progression has first term \(a\) and common ratio \(- 2 r\). The two progressions have the same sum to infinity. Find the value of \(r\).
The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first \(n\) terms. Find the value of \(n\).