Mean and median of the areas of enlarged rectangular cards forming a geometric sequence

A set of eight rectangular cards, \(C_1, C_2, \dots, C_8\), is cut from coloured paper. Card \(C_1\) is \(3\) cm wide and \(6\) cm tall. For \(1\leq n\leq7\), each side of card \(C_{n+1}\) is \(\frac43\) times the corresponding side of card \(C_n\).

1. 1. Show that card \(C_n\) has area \(18\left(\frac{16}{9}\right)^{n-1}\text{ cm}^2\).
   2. Hence, find the arithmetic mean of the areas of all eight cards, expressing it as \(p\left(\left(\frac{16}{9}\right)^{a}-1\right)\text{ cm}^2\), where \(p\in\mathbb{Q}^+\) and \(a\in\mathbb{Z}^+\).[5]
2. Find the median of the eight areas, expressing it as \(q\left(\frac{16}{9}\right)^3\text{ cm}^2\), where \(q\in\mathbb{Q}^+\).[3]