2021 RI P2 Q2

Three consecutive terms of a decreasing geometric progression has a product of \(5832\). If the first number is reduced by \(24\), these \(3\) numbers in the same order will form an arithmetic progression. Find the three terms of the geometric progression.[5]

The fractal called Sierpiński Triangle is depicted below. Fig. \(1\) shows an equilateral triangle of side \(1\). In stage \(1\), the triangle in Fig. \(1\) is divided into four smaller identical equilateral triangles and the middle triangle is removed to give the triangle shown in Fig. \(2\). In stage \(2\), the remaining three equilateral triangles in Fig. \(2\) are each divided into four smaller identical equilateral triangles and the middle triangles are removed to give the triangle shown in Fig. \(3\) and the process continues.

Let \({{T}_{n}}\) be the total area of triangles removed after \(n\) stages of the process.