If \(a_r\) is the coefficient of \(x^r\) in the expansion of \((1+x+x^2)^n\) in a series of ascending powers of \(x\), prove that
If £\(P\) is the present value of an annuity of £\(A\), to continue for \(n\) years, at \(100r\) per cent. per annum compound interest, prove that \[ \frac{Pr}{A} = 1-(1+r)^{-n}. \] If £\(Q\) is the present value of an annuity of £1 on the life of a man, shew that in order to receive £\(R\) at his death the payment to be made immediately and repeated annually is \[ £\frac{R(1-Qr)}{Q(1+r)}. \]
Find the number of combinations of \(m\) unlike things \(r\) at a time. Prove that the number of combinations \(n\) at a time of \(2n\) things, of which \(n\) are alike and the rest all different, is \(2^n\).
If \[ \frac{\cos(\alpha-3\theta)}{\cos^3\theta} = \frac{\sin(\alpha-3\theta)}{\sin^3\theta} = m, \] prove that \[ m^2+m\cos\alpha=2. \]
A gun is fired from a fort \(A\), and the intervals between seeing the flash and hearing the report at two stations \(B,C\) are \(t,t'\) respectively. \(D\) is a point in \(BC\) produced at a known distance \(a\) from \(A\). Prove that if \(BD=b\) and \(CD=c\), the speed of sound is \[ \left\{\frac{(b-c)(a^2-bc)}{bt'^2-ct^2}\right\}^{\frac{1}{2}}. \] Examine the case when \(a^2=bc\).
In a triangle \(ABC\), with the usual notation, prove that \[ r=4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}. \] A circle is drawn touching the circumscribed circle of the triangle internally, and also touching \(AB, AC\). Prove that its radius is \[ r\sec^2\frac{A}{2}. \]
If \[ \tan\alpha = \cos2\omega\cdot\tan\lambda, \] prove that \[ \lambda-\alpha = \tan^2\omega\cdot\sin2\alpha + \frac{1}{2}\tan^4\omega\cdot\sin4\alpha + \frac{1}{3}\tan^6\omega\cdot\sin6\alpha + \dots \]
Express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that the values of \(x\) which satisfy the equation \[ 1-nx - \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + (-1)^{\frac{1}{2}n(n-1)}x^n=0 \] are given by \[ x=\tan\frac{(4r+1)\pi}{4n}, \] where \(r\) is any integer.
Assuming that \(x\{\log(1+x)\}^{-1}\) can be expanded in ascending powers of \(x\), find the first four terms in the expansion. Hence show that a capital sum accumulating at compound interest at \(r\) per cent. per annum will be increased tenfold after \(\left(\frac{230.26}{r}+1.15\right)\) years.
Sum the series: