(a) The common difference \(d\) of the arithmetic progression is given by:

\(d = (2p - 6) - \frac{p^2}{6} = p - (2p - 6)\)

Equating the two expressions for \(d\):

\(2(2p - 6) = p + \frac{p^2}{6}\)

\(\Rightarrow p^2 - 18p + 72 = 0\)

Solving this quadratic equation by factorization:

\((p - 6)(p - 12) = 0\)

Since \(d \neq 0\), \(p = 12\).

(b) For the geometric progression, the common ratio \(r\) is:

\(r = \frac{2p - 6}{\frac{p^2}{6}} = \frac{18}{24} = \frac{3}{4}\)

The sum to infinity \(S\) is given by:

\(S = \frac{24}{1 - \frac{3}{4}} = 96\)