Shew that, if \(a_1, a_2 \dots a_n\) are unequal positive numbers, then \[ \frac{a_1+a_2+\dots+a_n}{n} > (a_1 a_2 \dots a_n)^{1/n} \] and, if also, \(a_1+a_2+\dots+a_n < 1\) then \((1+a_1)(1+a_2)\dots(1+a_n) < [1-(a_1+a_2+\dots+a_n)]^{-1}\).
(i) Sum to \(m\) terms the series whose \(n\)th term is \[ (a+\overline{n-1}b)(a+nb)\dots(a+\overline{n+r-2}b). \] (ii) Prove that \[ \frac{1^2}{n-1}\frac{1}{n+1} + \frac{2^2}{n-2}\frac{1}{n+2} + \dots + \frac{(n-1)^2}{1}\frac{1}{2n-1} + \frac{n^2}{2n} = \frac{2^{2n-3}}{2n-1}. \]
Find from first principles the differential coefficients of (i) \(\sin x\), (ii) \(\log_e(1+x^2)\). Find the value of \(\dfrac{du}{dx}\) when \(u = \log\dfrac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1} + 2\tan^{-1}\dfrac{\sqrt{2}x}{1-x^2}\).
State Maclaurin's Theorem for the expansion of \(f(x)\). Apply this method to the expansion of \(\sin\left(x+\dfrac{\pi}{4}\right)\) in ascending powers of \(x\).
If \(e^{x-y^2} = x-y\), prove that \(y^2\dfrac{\partial z}{\partial x} + x\dfrac{\partial z}{\partial y} = x^2-y^2\), and also that \[ y^2\frac{\partial^2 z}{\partial x^2} + x^2\frac{\partial^2 z}{\partial y^2} + 2xy\frac{\partial^2 z}{\partial x \partial y} + x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 0. \] % There seems to be a typo in the question, z is not defined. Assuming z = x-y.
Find the equations to the tangent and normal to the curve \(y=f(x)\) at any point. A circle is described passing through the origin and touching the curve \[ ax^3 = (x^2+y^2)y^2 \] at the point \((x,y)\). Shew that the circle cuts the axis of \(x\) also at the point whose abscissa is \(3x^2a/y^2\).
Prove the formula for the radius of curvature \(\rho=r\dfrac{dr}{dp}\). At any point of a rectangular hyperbola prove that \(3\rho\dfrac{d^2 p}{ds^2} - 2\left(\dfrac{dp}{ds}\right)^2\) is constant.
If \[ E(m) = 1+m+\frac{m^2}{2!} + \dots + \frac{m^r}{r!} + \dots, \] prove that \[ E(m) \times E(n) = E(m+n). \] Hence shew that \(E(x) = \{E(1)\}^x\) for all real values of \(x\).
Prove that the geometric mean between two quantities is also the geometric mean between their arithmetic and harmonic means. Sum the series \[ a+(a+b)r+(a+2b)r^2+(a+3b)r^3+\dots+(a+\overline{n-1}b)r^{n-1}. \]
Prove that, if \(p\) is a prime number, and \(x\) is any number less than \(p\) except \(1\) and \(p-1\), then another such number \(y\) can be found so that \(xy \equiv 1 \pmod p\). Hence prove that \((p-1)!+1 \equiv 0 \pmod p\).