Answer only one of the following two alternatives.

EITHER

\begin{enumerate}[label=(\roman*)]
\item By considering $(2r + 1)^2 - (2r - 1)^2$, use the method of differences to prove that
$$\sum_{r=1}^n r = \frac{1}{2}n(n + 1).$$ [3]

\item By considering $(2r + 1)^4 - (2r - 1)^4$, use the method of differences and the result given in part (i) to prove that
$$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n + 1)^2.$$ [5]
\end{enumerate}

The sums $S$ and $T$ are defined as follows:
$$S = 1^3 + 2^3 + 3^3 + 4^3 + \ldots + (2N)^3 + (2N + 1)^3,$$
$$T = 1^3 + 3^3 + 5^3 + 7^3 + \ldots + (2N - 1)^3 + (2N + 1)^3.$$

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Use the result given in part (ii) to show that $S = (2N + 1)^2(N + 1)^2$. [1]

\item Hence, or otherwise, find an expression in terms of $N$ for $T$, factorising your answer as far as possible. [2]

\item Deduce the value of $\frac{S}{T}$ as $N \to \infty$. [2]
\end{enumerate}

OR

The curve $C$ has equation
$$x^2 + 2xy = y^3 - 2.$$

\begin{enumerate}[label=(\roman*)]
\item Show that $A(-1, 1)$ is the only point on $C$ with $x$-coordinate equal to $-1$. [2]
\end{enumerate}

For $n \geqslant 1$, let $A_n$ denote the value of $\frac{d^n y}{dx^n}$ at the point $A(-1, 1)$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show that $A_1 = 0$. [3]

\item Show that $A_2 = \frac{2}{5}$. [3]
\end{enumerate}

Let $I_n = \int_{-1}^0 x^n \frac{d^n y}{dx^n} dx$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Show that for $n \geqslant 2$,
$$I_n = (-1)^{n+1} A_{n-1} - nI_{n-1}.$$ [3]

\item Deduce the value of $I_3$ in terms of $I_1$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2018 Q11 [26]}}