\(\frac{2k - 1}{k + 1} = \frac{3k + 1}{2k - 1}\)

\((k + 1)(3k + 1) = (2k - 1)(2k - 1)\)

\(3k^{2} + 4k + 1 = 4k^{2} - 4k + 1\)

\(4k^{2} - 3k^{2} - 4k - 4k + 1 - 1 = 0\)

\(k^{2} - 8k = 0\)

\(k(k - 8) = 0\)

\(\therefore \text{k = 0 or 8}\)

The terms of the sequence given k = 0: (1, -1, 1)

\(\implies \text{The common ratio r = -1}\)

The terms of the sequence given k = 8: (9, 15, 25)

\(\implies \text{The common ratio r = } \frac{5}{3}\)

The possible values of the common ratio are -1 and \(\frac{5}{3}\).