2025 May TZ1 P1 Q6

Consider a sequence of ten rectangular picture frames \(F_1, F_2, \dots, F_9, F_{10}\).
Picture frame \(F_1\) has width \(4\) cm and height \(5\) cm.
The width and height of picture frame \(F_n\), are each increased by \(50\%\) to generate the width and height of the next picture frame \(F_{n+1}\), for \(n \in \mathbb{Z}^+\), \(1 \leq n \leq 9\).

1. [5]

   1. Show that the area of picture frame \(F_n\) is \(20 \left(\frac{9}{4} \right)^{n-1}\) cm\(^2\).

   2. Hence, find the mean area of the ten picture frames, giving your answer in the form \(p \left( \left(\frac{9}{4} \right)^a - 1\right)\) cm\(^2\), where \(p \in \mathbb{Q}^+, a \in \mathbb{Z}^+\).

2. Find the median area of the ten picture frames, giving your answer in the form \(q \left(\frac{9}{4}\right)^4\) cm\(^2\), where \(q \in \mathbb{Q}^+\).

   [3]