(a)[3]
One geometric progression begins with first term $3a$ and has common ratio $r$. A second geometric progression begins with first term $a$ and has common ratio $-2r$. These two progressions have equal sums to infinity. Determine the value of $r$.
(b)[3]
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 identical sums for the first $n$ terms. Determine the value of $n$.