(i) For a geometric progression, the ratio between consecutive terms is constant. Therefore, \(\frac{5k - 6}{3k} = \frac{6k - 4}{5k - 6}\).

Cross-multiplying gives \((5k - 6)^2 = 3k(6k - 4)\).

Expanding both sides: \(25k^2 - 60k + 36 = 18k^2 - 12k\).

Rearranging gives \(7k^2 - 48k + 36 = 0\).

(ii) Solving \(7k^2 - 48k + 36 = 0\) gives \(k = \frac{6}{7}\) or \(k = 6\).

When \(k = \frac{6}{7}\), the first term is \(3 \times \frac{6}{7} = \frac{18}{7}\) and the common ratio \(r = \frac{2}{3}\).

When \(k = 6\), the common ratio \(r = \frac{4}{3}\).

(iii) The progression is convergent if \(|r| < 1\). Thus, \(r = \frac{2}{3}\) is convergent.

The sum to infinity is \(S_\infty = \frac{a}{1-r}\) where \(a = \frac{18}{7}\) and \(r = \frac{2}{3}\).

\(S_\infty = \frac{\frac{18}{7}}{1 - \frac{2}{3}} = \frac{18}{7} \times 3 = \frac{54}{35}\) or 1.54.