Solve the equations:
Show how to determine \(u_n\) from the equation \[ Au_n+Bu_{n+1}+Cu_{n+2}=0, \] where \(A, B, C\) are constants, when the values of \(u_0\) and \(u_1\) are given; and show that the complete solution of \[ u_n - 2u_{n+1}\cos\theta+u_{n+2}=0 \] is of the form \(H\cos n\theta+K\sin n\theta\).
If \(AB, BC, CD\) are three sides of a quadrilateral of lengths \(a,b,c\) respectively, and if \(\angle ABC=\theta, \angle BCD=\phi\), and the angle between \(AB\) and \(DC\) produced is \(\psi\), prove that \[ AD^2 = a^2+b^2+c^2 - 2ab\cos\theta - 2bc\cos\phi - 2ca\cos\psi. \]
If \(\alpha+\beta+\gamma+\delta=2\pi\), show that \[ (\sin 2\alpha+\sin 2\beta+\sin 2\gamma+\sin 2\delta)^2 = (\cos\beta.\cos\gamma-\cos\alpha.\cos\delta)(\cos\gamma.\cos\alpha-\cos\beta.\cos\delta)(\cos\alpha.\cos\beta-\cos\gamma.\cos\delta). \] % Note: there appears to be a factor missing from the original scan. This is a common identity for the area of a cyclic quadrilateral, but it's usually written with a factor of 16 or 64. I will transcribe as written.
The angles of elevation of the top of a mountain from three points \(A, B, C\) in a base line are observed to be \(\alpha, \beta, \gamma\) respectively. Prove that the height of the mountain is \[ (-AB \cdot BC \cdot CA)^{\frac{1}{2}}(BC\cot^2\alpha+CA\cot^2\beta+AB\cot^2\gamma)^{-\frac{1}{2}}, \] where regard is paid to the sense of the lines.
Find the sum \(s_n\) of \(n\) terms of the series \[ \sin x + \sin 2x + \sin 3x + \dots, \] and prove that \[ \text{Limit}_{n\to\infty} \frac{s_1+s_2+\dots+s_n}{n} = \tfrac{1}{2}\cot\tfrac{1}{2}x. \]
If \(A=a^2(a+b+c)+3abc\), \(B=b^2(a+b+c)+3abc\) and \(C=c^2(a+b+c)+3abc\), where \(ab+bc+ca=0\), then \((AB+BC+CA)abc = ABC\).
Solve the equations \begin{align*} a_1x+b_1y+c_1z &= d_1, \\ a_2x+b_2y+c_2z &= d_2, \\ a_3x+b_3y+c_3z &= d_3. \end{align*} And eliminate \(a, b, c\) from the equations \[ x = \frac{a}{b+c}, \quad y=\frac{b}{c+a}, \quad z=\frac{c}{a+b}. \]
(i) Prove with the usual notation that \({}^nC_r = \frac{n}{r}{}^{n-1}C_{r-1}\) and derive the number of combinations of \(n\) letters \(r\) at a time. (ii) Shew that the number of parts into which a plane is divided by \(n\) straight lines no two of which are parallel and no three concurrent is \(\frac{1}{2}(n^2+n+2)\).
Prove that, if \(p\) and \(q\) are positive integers, \(e^{p/q} = 1 + \dfrac{p}{q} + \dfrac{p^2}{2q^2} + \dfrac{p^3}{3!q^3} + \dots\). If \(u = 1 + \dfrac{1}{1.3} + \dfrac{1}{1.3.5} + \dfrac{1}{1.3.5.7} + \dots\), and \(v = 1 - \dfrac{1}{1}\cdot\dfrac{1}{3} + \dfrac{1}{4}\cdot\dfrac{1}{2\cdot 5} - \dfrac{1}{8}\cdot\dfrac{1}{3\cdot 7} + \dfrac{1}{16}\cdot\dfrac{1}{4\cdot 9} - \dots\), prove that, to four places of decimals, \(u^2/v^2=e\).