6.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-22_481_1139_189_463}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Figure 2 shows the first few iterations in the construction of a curve, $L$.\\
Starting with a straight line $L _ { 0 }$ of length 4 , the middle half of this line is replaced by three sides of a trapezium above $L _ { 0 }$ as shown, such that the length of each of these sides is $\frac { 1 } { 4 }$ of the length of $L _ { 0 }$

After the first iteration each line segment has length one.\\
In subsequent iterations, each line segment parallel to $L _ { 0 }$ similarly has its middle half replaced by three sides of a trapezium above that line segment, with each side $\frac { 1 } { 4 }$ the length of that line segment.

Line segments in $L _ { n }$ are either parallel to $L _ { 0 }$ or are sloped.
\begin{enumerate}[label=(\alph*)]
\item Show that the length of $L _ { 2 }$ is $\frac { 23 } { 4 }$
\item Write down the number of
\begin{enumerate}[label=(\roman*)]
\item line segments in $L _ { n }$ that are parallel to $L _ { 0 }$
\item sloped line segments in $L _ { 2 }$ that are not in $L _ { 1 }$
\item new sloped line segments that are created by the ( $n + 1$ )th iteration.
\end{enumerate}\item Hence find the length of $L _ { n }$ as $n \rightarrow \infty$

The area enclosed between $L _ { 0 }$ and $L _ { n }$ is $A _ { n }$
\item Find the value of $A _ { 1 }$
\item Find, in terms of $n$, an expression for $A _ { n + 1 } - A _ { n }$
\item Hence find the value of $A _ { n }$ as $n \rightarrow \infty$

The same construction as described above is applied externally to the three sides of an equilateral triangle of side length $a$.\\
Given that the limit of the area of the resulting shape is $26 \sqrt { 3 }$
\item find the value of $a$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2022 Q6 [24]}}