5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-4_493_1324_279_367}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows part of a sequence $S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots$, of model snowflakes. The first term $S _ { 1 }$ consists of a single square of side $a$. To obtain $S _ { 2 }$, the middle third of each edge is replaced with a new square, of side $\frac { a } { 3 }$, as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square $\frac { 1 } { 3 }$ of the size, as illustrated by $S _ { 3 }$ in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Deduce that to form $S _ { 4 } , 36$ new squares of side $\frac { a } { 27 }$ must be added to $S _ { 3 }$.
\item Show that the perimeters of $S _ { 2 }$ and $S _ { 3 }$ are $\frac { 20 a } { 3 }$ and $\frac { 28 a } { 3 }$ respectively.
\item Find the perimeter of $S _ { n }$.
\item Describe what happens to the perimeter of $S _ { n }$ as $n$ increases.
\item Find the areas of $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$.
\item Find the smallest value of the constant $S$ such that the area of $S _ { n } < S$, for all values of $n$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588}

Figure 2 shows a sketch of the curve $C$ with equation $y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }$.\\
The point $P$ on $C$ has coordinates $\left( x , \tan \frac { x } { 2 } \right)$.\\
The vertices of rectangle $R$ are at $( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)$ and $\left( x , \tan \frac { x } { 2 } \right)$ as shown in Figure 2.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2007 Q5 [15]}}