(a) For the arithmetic progression, the second term is \(\frac{3}{2}a\) and the third term is \(b\). The common difference \(d\) is given by:

\(d = \frac{3}{2}a - a = \frac{a}{2}\)

\(b = \frac{3}{2}a + \frac{a}{2} = 2a\)

For the geometric progression, the second term is 18 and the third term is \(b + 3\). The common ratio \(r\) is given by:

\(r = \frac{18}{a}\) and \(b + 3 = 18r\)

Substituting \(b = 2a\) into \(b + 3 = 18r\), we get:

\(2a + 3 = 18 \times \frac{18}{a}\)

\(2a^2 + 3a - 324 = 0\)

Solving this quadratic equation, we find \(a = 12\) and \(b = 24\).

(b) The common difference \(d = 6\) (since \(d = \frac{a}{2} = 6\)). The sum of the first 20 terms \(S_{20}\) is given by:

\(S_{20} = \frac{20}{2} \times (2 \times 12 + 19 \times 6)\)

\(S_{20} = 10 \times (24 + 114) = 10 \times 138 = 1380\)