Find the sum of the series \(a_0+a_1x+a_2x^2+\dots\), whose coefficients satisfy the relation \[ 3a_n - 7a_{n-1} + 5a_{n-2} - a_{n-3} = 0, \text{ and } a_0=1, a_1=8, a_2=17 \] proving that \[ 2a_n = 20n - 7 + 3^{2-n}. \]
Find the only value of \(x\) which satisfies the equation \[ 3\tan^{-1}x + \tan^{-1}3x = \frac{1}{4}\pi, \] where \(\tan^{-1}x\) and \(\tan^{-1}3x\) each lie between \(0\) and \(\frac{1}{2}\pi\).
An aeroplane is travelling in a horizontal line with constant velocity. Explain how two observers at the same level and at a known distance apart may calculate the speed and direction of motion when two pairs of simultaneous observations of the bearing of the aeroplane are made at observed times.
The real quantities \(x,y,u,v\) are connected by the equation \[ \cosh(x+iy) = \cot(u+iv). \] Prove that \[ \frac{\sinh 2y}{\sin 2u} = -\tanh x \tan y, \] and that \[ \coth 2v = -(\cosh 2x + \cos 2y + 2)/4 \sinh x \sin y. \]
Expand \((x^2+1)^{\frac{1}{2}}\sinh^{-1}x\) in a series of ascending powers of \(x\), and if \(a_n\) is the coefficient of \(x^n\), prove that \[ a_{n+2} = -\left(\frac{n-1}{n+2}\right)a_n. \quad (n>1.) \]
Prove that if \(s\) is the arc of the curve \(3ay^2 = x(x-a)^2\) from the origin to the point \((x,y)\), then \[ s^2=y^2+\frac{4}{9}x^2. \] If \(S\) is the whole length of the loop of the curve, \(A\) its area and \(B\) its greatest breadth parallel to the \(y\) axis, prove that \(A=\frac{4}{15}BS\).
Evaluate \[ \int \frac{dx}{\sqrt{(1+\sin x)(2-\sin x)}}. \] Prove that \[ \int_0^1 \frac{(4x^2+3)dx}{8x^2+4x+5} = \frac{1}{4} - \frac{1}{8}\log\frac{17}{5} + \frac{1}{4}\tan^{-1}\frac{6}{7}, \] and \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1+2\cos x} = \frac{1}{\sqrt{3}}\log(2+\sqrt{3}). \]
Prove that \[ \sin A + \sin B + \sin C - \sin(A+B+C) = 4\sin\frac{1}{2}(B+C)\sin\frac{1}{2}(C+A)\sin\frac{1}{2}(A+B). \] If \(A+B+C=180^\circ\), shew that \begin{align*} \cos 3A \sin(B-C) + \cos 3B \sin(C-A) + \cos 3C \sin(A-B) \\ + 4\sin(B-C)\sin(C-A)\sin(A-B) = 0. \end{align*}
If \(R\) and \(r\) are the radii of the circumscribed and inscribed circles of a triangle \(ABC\), prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] Show also that the radius of the circle inscribed in the triangle formed by joining the centres of the three escribed circles is \[ \frac{4R\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}}{\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}}. \]
Find the sum of \(n\) terms of the series: