5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d269bf1-f481-46bd-b9d3-fea211b186cf-14_317_1557_255_255}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the first three stages of a pattern that is created by a recursive process.
The process starts with a square and proceeds as follows
- each square is replaced by 5 smaller squares each \(\frac { 1 } { 9 }\) th the size of the square being replaced
- the 5 smaller squares are the ones in each corner and the one in the centre
- once each of the squares has been replaced, the square immediately to the right and above the centre square of the pattern is then removed
Let \(u _ { n }\) be the number of squares in the pattern in stage \(n\), where stage 1 is the original square.
- Explain why \(u _ { n }\) satisfies the recurrence system
$$u _ { 1 } = 1 \quad u _ { n + 1 } = 5 u _ { n } - 1 \quad ( n = 1,2,3 , \ldots )$$
- Solve this recurrence system.
Given that the initial square has area 25
- determine the total area of all the squares in stage 8 of the pattern, giving your answer to 2 significant figures.