Sum the series \[ n^2+2(n-1)^2+3(n-2)^2+\dots, \] where \(n\) is a positive integer. Prove that \[ S_1 + \frac{S_2}{2!} + \dots + \frac{S_n}{n!} + \dots \text{ to infinity} = \frac{17e}{6}, \] where \(S_n\) is the sum of the first \(n\) positive integers.
Prove that the roots of the equation \[ x^3+3px^2+3qx+r=0 \] will be in geometrical progression if \[ p^3r=q^3; \] but will be in harmonical progression if \[ 2q^3 = r(3pq-r). \]
Prove that the number of homogeneous products of \(r\) dimensions which can be formed with \(n\) letters, where each letter may be repeated any number of times, is \[ \frac{(n+r-1)!}{r!(n-1)!}. \] An examination consists of three papers, to each of which \(m\) marks are assigned as a maximum. One candidate obtains a total of \(2m\) marks on the three papers. Shew that there are \(\frac{1}{2}(m+1)(m+2)\) ways in which this may occur.
Prove that as the real variable \(x\) changes steadily from \(-\infty\) to \(+\infty\), the function
\[ y=\frac{(x-a)^2}{x-b} \]
(where \(a\) and \(b\) are real and \(a
If \[ (a+b)\tan(\theta-\phi) = (a-b)\tan(\theta+\phi) \] and \[ a\cos 2\phi + b\cos 2\theta = c, \] prove that \[ a^2-b^2+c^2=2ac\cos 2\phi. \]
From a point \(P\) on the plane sloping face of a hill two straight paths \(PQ\) and \(PR\) are drawn to the horizontal road \(QR\) at the foot of the hill. \(PQ\) lies in a vertical plane due south, and \(PR\) is a vertical plane due east. \(PQ=a, PR=b\), and the angle \(QPR\) is \(\alpha\). Prove that the slope of the hill to the horizontal is \[ \sin^{-1}\left( \frac{(a^2+b^2-2ab\cos\alpha)^{\frac{1}{2}}}{ab\sin\alpha\tan\alpha} \right). \]
\(I_1, I_2, I_3\) are the centres of the escribed circles of the triangle \(ABC\). With the usual notation, prove that the radius of the circle \(I_1 I_2 I_3\) is \(2R\), and that the area of the triangle \(I_1 I_2 I_3\) is \(\dfrac{abc}{2r}\).
Prove that \[ \sum_{p=1}^{p=n} \sin\frac{2p\pi}{n} = \sum_{p=1}^{p=n} \cos\frac{2p\pi}{n} = 0, \] if \(n\) is a positive integer \(>1\). Prove that all the real values of \(\theta\) between \(0\) and \(\frac{\pi}{2}\) which satisfy the equation \[ \tan(\cot\theta) = \cot(\tan\theta) \] are \[ \sin^{-1}\frac{4}{(2r+1)\pi}, \] where \(r\) is any positive integer.
If \[ (1+x)^n = c_0+c_1 x + c_2 x^2 + \dots \] prove that \[ c_0-c_2+c_4-\dots = 2^{\frac{n}{2}}\cos\frac{1}{4}n\pi. \] \(A_1 A_2 A_3 \dots A_n\) is a regular polygon of \(n\) sides inscribed in a circle of radius \(a\). \(P\) is any point on the circumference, \(O\) is the centre, and \(\angle POA_1=\theta\). Prove that \[ PA_1 \cdot PA_2 \cdot PA_3 \dots PA_n = 2a^n \sin\frac{1}{2}n\theta. \]
State the laws of friction and explain what is meant by the angle of friction. A uniform rod rests in limiting equilibrium with its ends on a rough circular band whose plane is vertical. Prove that its inclination (\(\theta\)) to the vertical is given by the equation \[ \sin\lambda\sin(\theta+\lambda) = \cos\theta\cos^2\alpha, \] where \(\lambda\) is the angle of friction and \(2\alpha\) the angle subtended by the rod at the centre of the circle.