Let the first term of the geometric progression (GP) be \(a = 64\) and the common ratio be \(r\). The second term is \(ar = 48\), so \(64r = 48\), giving \(r = \frac{3}{4}\).

The third term of the GP is \(ar^2 = 64 \times \left(\frac{3}{4}\right)^2 = 36\).

For the arithmetic progression (AP), the first term \(a = 64\) and the ninth term \(a + 8d = 48\). Solving for \(d\), we have:

\(64 + 8d = 48\)

\(8d = 48 - 64 = -16\)

\(d = -2\)

The nth term of the AP is \(a + (n-1)d = 36\). Substituting the known values:

\(64 + (n-1)(-2) = 36\)

\(64 - 2(n-1) = 36\)

\(64 - 2n + 2 = 36\)

\(66 - 2n = 36\)

\(2n = 66 - 36 = 30\)

\(n = 15\)