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Re: HSC 2013 4U Marathon Yep that is pretty much it.

Re: MX2 Integration Marathon Interesting method, nice work. Alternatively, if we consider the integral to be I_n We find that I(n-1)-I(n) = (something integratable) Then take a sum of both sides from k=1 to n, the LHS telescopes to leave I(1) and I(n), the RHS being an unevaluatable sum...

Re: MX2 Integration Marathon Ah upper limit is supposed to be 1/2, not 1 sorry about that.

Re: HSC 2013 4U Marathon $From a set of$ \ \ r \ \ $even integers$ \ \ x_1, x_2, \dots , x_r \ \ $and$ \ \ s \ \ $odd integers$ \ \ y_1, y_2, \dots , y_s $Consider the set$ \ \left \{ (-1)^{x_1}, (-1)^{x_2} , (-1)^{x_3}, \dots , (-1)^{x_r} , (-1)^{y_1} , (-1)^{y_2} , \dots , (-1)^{y_s} \right...

Re: MX2 Integration Marathon $For some non-negative integer$ \ \ n \ \ $evaluate$ \int_0^{1/2} \frac{x^{2n}}{1-x^2} \ dx

Re: HSC 2013 3U Marathon Thread Well done. That is the method I had in mind. $The sequence$ \ \ u_1, u_2, u_3, \dots, u_n \ \ \dots $Is an arithmetic sequence that consists of only positive numbers$ $Show that$ \sum_{k=2}^{n} \frac{1}{\sqrt{u_{n}} + \sqrt{u_{n-1}}} =...

Re: MX2 Integration Marathon Ok: \int \frac{dx}{\sin^3 x + \cos^3 x} = \int \frac{dx}{(\sin x + \cos x)(1-\sin x \cos x} = \int \frac{\cos x - \sin x}{(\cos^2 x - \sin^2 x)(1-\sin x \cos x)} u = \cos x + \sin x du = \cos x - \sin x u^2 - 1 = \sin 2x Make a right angled triangle, we...

Re: HSC 2013 4U Marathon You're first part is correct, for the second though, I'd advise against the case by case basis, it might work though. There is an easier method however.

Re: HSC 2013 3U Marathon Thread $A projectile with a velocity$ \ \ V \ \ $at an angle$ \ \ \theta \ \ $to the horizontal$ $It lands on an inclined plane at an angle$ \ \ \alpha \ \ $to the horizontal as shown in the diagram$ $Prove that maximum range achieved can only be done so if$...

Re: HSC 2013 4U Marathon $A sequence is defined with the recurrence formula$ F_{n+2} = F_n + F_{n+1} \ \ \ n\geq 0 F_0 = 0 F_1 = 1 $i) Prove with mathematical induction$ \ \ F_n F_{n+3} - F_{n+1}F_{n+2} = (-1)^{n+1} $ii) Hence prove that$ \tan^{-1} \left(\frac{1}{F_{2r+1}} \right ) +...

Re: HSC 2013 4U Marathon Z_n(\alpha) = \sum_{k=1}^{n}\frac{1}{k^{\alpha}} $Prove that for$ \ \ \alpha > 1 \lim_{n \to \infty} Z_n (\alpha) = L \ \ $for some value$ \ L $Also prove that if$ \ \ \alpha \leq 1 Z_n(\alpha) \to \infty \ \ \ $as$ \ n \ $increases without bound$

Re: HSC 2013 4U Marathon $Let$ \ LHS= P RHS = Q \frac{P(m+1) - a^m}{m} = Q m(P-Q) = a^m - P P-Q = \frac{a^m - P}{m} $Since$ \ \ a > 1 P = \frac{a^m + a^{m-1} + \dots + 1}{m+1} < \frac{a^m + a^m + \dots + a^m}{m+1} = a^m \therefore P < a^m \therefore P - Q > 0 \therefore P > Q

Re: HSC 2013 4U Marathon Hmm, I'm not sure if your third line is always true.

Re: HSC 2013 4U Marathon $Prove that$ \frac{a^{m} + a^{m-1} + \dots + 1}{m+1} > \frac{a^{m-1} + a^{m-2} + \dots + 1}{m} $For some real$ a> 1

Re: HSC 2013 4U Marathon a^2+b^2+c^2+d^2 - ab-bc-cd-d+ 2/5 = 0 \Rightarrow \ \ \left(a - \frac{b}{2} \right)^2 + \frac{3}{4} \left(b - \frac{2c}{3}\right)^2 + \frac{2}{3} \left( c- \frac{3d}{4} \right)^2 + \frac{5}{8} \left( d - \frac{4}{5} \right)^2 = 0 So just completing the square gives...

Re: HSC 2013 4U Marathon Oh yes real solutions definitely.

Re: HSC 2013 4U Marathon Nope, its supposed to be d, there is no real significance in the number and the apparent symmetry though.

Re: HSC 2013 4U Marathon $i) Show that$ \ \ \cos(n\theta) \cos \theta = \frac{1}{2}(\cos(n+1)\theta + \cos(n-1)\theta) \ \ \ \fbox{1} $ii) A sequence of polynomials$ \ \ C_n \ \ $is defined by$ C_n(\cos \theta) = \cos(n\theta) $Prove the recurrence formulae$ C_0(x) = 1 C_1(x) = x...

Re: MX2 Integration Marathon This is a great integral, a little long and the answer isn't very elegant, but the path towards it is I think. \int \frac{dx}{\sin^3 x + \cos^3 x}

Re: HSC 2013 4U Marathon $Solve the equation$ a^2+b^2+c^2+d^2 - ab - bc -cd - d + \frac{2}{5} = 0