Problems

(i) Sum to \(n\) terms the series \[ \frac{1}{1.3.5} + \frac{2}{3.5.7} + \frac{3}{5.7.9} + \dots. \] (ii) Prove that \[ \frac{2^3}{2!} + \frac{3^3}{3!} + \frac{4^3}{4!} + \dots \text{ to infinity} = 5e-1. \]

If \(p\) is small, so that \(p^3\) is negligible, prove that an approximation to a solution of the equation \(x^{2+p}=a^2\) is \[ x = a - \frac{1}{2}ap\log_e a + \frac{1}{8}ap^2(2+\log_e a)\log_e a. \]

If \(\theta\) and \(\phi\) are unequal and less than \(180^{\circ}\), and if \[ (x-a)\cos 2\theta + y\sin 2\theta = (x-a)\cos 2\phi + y\sin 2\phi = a, \] and \[ \tan\theta - \tan\phi = 2e, \] prove that \[ y^2=2ax-(1-e^2)x^2. \]

\(APQB\) is a straight line, and the lengths of \(AQ, PB\) and \(AB\) are \(2a, 2b\) and \(2c\) respectively. Circles are described on diameters \(AQ, PB, AB\). Prove that the radius of a circle that touches these three circles is \[ \frac{c(c-a)(c-b)}{c^2-ab}. \]

A regular polygon of \(n\) sides is inscribed in a circle of radius \(a\), and from any point \(P\) on the circumference chords are drawn to the angular points; if these chords are of lengths \(c_1, c_2, c_3, \dots, c_n\), beginning with the chord drawn to the nearest angular point and taking the rest in order, find the value of \[ c_1c_2+c_2c_3+c_3c_4+\dots+c_{n-1}c_n+c_nc_1, \] and prove that it is the same for every point \(P\) on the circumference.

Sum to \(n\) terms the series

1. [(i)] \(\cos\alpha.\cos\alpha + \cos^2\alpha.\cos 2\alpha + \cos^3\alpha.\cos 3\alpha + \dots\),
2. [(ii)] \(\sec^2\frac{\pi}{4n} + \sec^2\frac{3\pi}{4n} + \sec^2\frac{5\pi}{4n} + \dots\).

Prove that \[ \tan^{-1}\frac{\tan 2\alpha+\tanh 2\beta}{\tan 2\alpha-\tanh 2\beta} + \tan^{-1}\frac{\tan\alpha-\tanh\beta}{\tan\alpha+\tanh\beta} = \tan^{-1}(\cot\alpha\coth\beta). \]

\(P, Q, R\) are any points on the sides \(BC, CA, AB\) respectively of the triangle \(ABC\). Prove that the circles \(AQR, BRP, CPQ\) meet in a point. \par Using a special case of this theorem, shew how to construct points \(L\) and \(M\) such that the angles \(LBC, LCA, LAB\) are equal, and the angles \(MCB, MAC, MBA\) are equal.

Find the angle between the lines given by the equation \[ ax^2+2hxy+by^2=0, \] and obtain the equation of the pair of lines bisecting the angles between the lines. Hence write down the equation of the most general pair of lines having the same bisectors.

Prove that the centre of the rectangular hyperbola which passes through four concyclic points \(A, B, C, D\) lies on the perpendicular drawn from the mid-point of the line joining any two of the four points to the line joining the remaining two points. \par Explain how the asymptotes can be constructed.