Answer only one of the following two alternatives.
**EITHER**
Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1]
Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2]
The sums \(S\), \(T\) and \(U\) are defined as follows:
\begin{align}
S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2.
\end{align}
Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5]
Hence
- describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
- prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]
**OR**
The curves \(C_1\) and \(C_2\) have polar equations
$$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$
respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).
- Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
- In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
- Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is
$$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]