(i) For a geometric progression, the common ratio is given by \(r = \frac{32}{36} = \frac{8}{9}\).

The sum to infinity for a geometric series is \(S_\infty = \frac{a}{1-r}\), where \(a = 36\) and \(r = \frac{8}{9}\).

Thus, \(S_\infty = \frac{36}{1 - \frac{8}{9}} = \frac{36}{\frac{1}{9}} = 36 \times 9 = 324\).

(ii) For an arithmetic progression, the common difference is \(d = 32 - 36 = -4\).

The sum of an arithmetic series is given by \(S_n = \frac{n}{2} (2a + (n-1)d)\).

Given \(S_n = 0\), we have:

\(0 = \frac{n}{2} (72 + (n-1)(-4))\)

\(0 = \frac{n}{2} (72 - 4n + 4)\)

\(0 = \frac{n}{2} (76 - 4n)\)

\(0 = n(38 - 2n)\)

\(0 = 38n - 2n^2\)

\(2n^2 = 38n\)

\(n(2n - 38) = 0\)

\(n = 0\) or \(n = 19\)

Since \(n = 0\) is not a valid solution, \(n = 19\).