(i) The general term of an arithmetic progression is given by \(a_n = a + (n-1)d\).

For the 5th term, \(n = 5\):

\(a_5 = a + 4d\).

For the 15th term, \(n = 15\):

\(a_{15} = a + 14d\).

(ii) The terms \(a\), \(a + 4d\), and \(a + 14d\) form a geometric progression.

Thus, \(\frac{a + 4d}{a} = \frac{a + 14d}{a + 4d}\).

Cross-multiplying gives:

\((a + 4d)^2 = a(a + 14d)\).

Expanding both sides:

\(a^2 + 8ad + 16d^2 = a^2 + 14ad\).

Simplifying gives:

\(8ad + 16d^2 = 14ad\).

\(16d^2 = 6ad\).

\(8d = 3a\).

Thus, \(3a = 8d\).

(iii) The common ratio \(r\) of the geometric progression is:

\(r = \frac{a + 4d}{a}\) or \(r = \frac{a + 14d}{a + 4d}\).

Using \(r = \frac{a + 4d}{a}\):

\(r = \frac{a + 4d}{a} = 1 + \frac{4d}{a}\).

Substituting \(3a = 8d\), we have \(a = \frac{8d}{3}\).

\(r = 1 + \frac{4d}{\frac{8d}{3}} = 1 + \frac{12d}{8d} = 1 + \frac{3}{2} = 2.5\).