(a) The first 3 terms of an arithmetic progression are \(3 \sin 2 x, 5 \sin 2 x, 7 \sin 2 x\). (i) Show that the sum to \(n\) terms of this arithmetic progression can be written in the form \(n(n+a) \sin 2 x\), where \(a\) is a constant. (ii) Given that \(x=\frac{2 \pi}{3}\), find the exact sum of the first 20 terms.

(b) The first 3 terms of a geometric progression are \(\ln 2 y, \quad \ln 4 y^{2}, \quad \ln 16 y^{4}\). (i) Find the \(n\)th term of this geometric progression.

(ii) Find the sum to \(n\) terms of this geometric progression, giving your answer in its simplest form.

(c) The first 3 terms of a different geometric progression are \(\left(2 w-\frac{1}{4}\right),\left(2 w-\frac{1}{4}\right)^{2},\left(2 w-\frac{1}{4}\right)^{3}\). Find the values of \(w\) for which this geometric progression has a sum to infinity.