Answer only one of the following two alternatives.

EITHER

(i) By considering \((2r+1)^2-(2r-1)^2\), use the method of differences to prove that

\(\sum_{r=1}^{n}r=\frac12n(n+1).\)

(ii) By considering \((2r+1)^4-(2r-1)^4\), use the method of differences and the result given in part (i) to prove that

\(\sum_{r=1}^{n}r^3=\frac14n^2(n+1)^2.\)

The sums \(S\) and \(T\) are defined as follows:

\(S=1^3+2^3+3^3+4^3+\cdots+(2N)^3+(2N+1)^3,\)

\(T=1^3+3^3+5^3+7^3+\cdots+(2N-1)^3+(2N+1)^3.\)

(iii) Use the result given in part (ii) to show that \(S=(2N+1)^2(N+1)^2\).

(iv) Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.

(v) Deduce the value of \(\dfrac ST\) as \(N\to\infty\).

OR

The curve \(C\) has equation \(x^2+2xy=y^3-2\).

(i) Show that \(A(-1,1)\) is the only point on \(C\) with \(x\)-coordinate equal to \(-1\).

For \(n\ge1\), let \(A_n\) denote the value of \(\dfrac{d^ny}{dx^n}\) at the point \(A(-1,1)\).

(ii) Show that \(A_1=0\).

(iii) Show that \(A_2=\dfrac25\).

Let \(I_n=\int_{-1}^{0}x^n\dfrac{d^ny}{dx^n}\,dx\).

(iv) Show that for \(n\ge2\),

\(I_n=(-1)^{n+1}A_{n-1}-nI_{n-1}.\)

(v) Deduce the value of \(I_3\) in terms of \(I_1\).