(i) For an arithmetic progression, the second term is given by:

\(a + d = 96\)

and the fourth term is:

\(a + 3d = 54\)

Subtract the first equation from the second:

\((a + 3d) - (a + d) = 54 - 96\)

\(2d = -42\)

\(d = -21\)

Substitute \(d = -21\) back into \(a + d = 96\):

\(a - 21 = 96\)

\(a = 117\)

(ii) For a geometric progression, the second term is:

\(ar = 96\)

and the fourth term is:

\(ar^3 = 54\)

Divide the second equation by the first:

\(\frac{ar^3}{ar} = \frac{54}{96}\)

\(r^2 = \frac{9}{16}\)

\(r = \frac{3}{4}\)

Substitute \(r = \frac{3}{4}\) back into \(ar = 96\):

\(a \times \frac{3}{4} = 96\)

\(a = 128\)