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Re: HSC 2013 4U Marathon $Define a polynomial $ \ \ C_n(x) \ \ $such that it behaves that$ C_n(\cos \theta) = \cos(n\theta) $i) Prove that the Polynomial satisfies the recurrence relation with the initial conditions$ \ \ \ \ \ \ \fbox{3} \begin{cases} C_0(x) = 1 \\ C_1(x)=x \\ C_n(x) =...

Re: HSC 2013 4U Marathon It's a bit harsh to make people try and make their own inequality and bounds AND squeeze theorem twice isn't it? :s

Re: HSC 2013 3U Marathon Thread Great question. Nice work. ===================== $Find$ \int \frac{1}{x^2+10x+50} \ dx

Re: HSC 2013 3U Marathon Thread $Utilising a suitable substitution, find$ \int \frac{1+2\sqrt{x}}{2\sqrt{x}(x+\sqrt{x})} \ dx

Re: HSC 2013 3U Marathon Thread This. I'm not silly enough to post a 4U question in a 3U thread. In fact that integral is 2U.

Re: HSC 2013 3U Marathon Thread $Prove that$ \ \ \ \lim_{\alpha \to 0}\int_{\alpha}^{x} \ln t \ dt = x\ln x -x

Re: HSC 2013 4U Marathon f_0(x)= \frac{1}{1-x} $let$ \ \ f_r (x) = f(f(f(\dots (f(x))\dots )) $Where there are r iterations of the function$ $So$ \ \ f_2(x) = f(f(f(x))) $Evaluate$ \ \ \ f_{2013} (\pi)

Re: HSC 2013 3U Marathon Thread I meant, prove your value :s

Re: HSC 2013 3U Marathon Thread I don't believe you.

Re: HSC 2013 4U Marathon Great question: First lets establish the identity \sin x = 2^r \cos(x/2) \cos (x/4) \cos(x/8) \dots \cos(x/2^r) \sin(x/2^r) Take the natural logarithm of both sides, ensuring that cos(x/2^k) is always positive (absolute value), and use logarithm laws...

If it helps other people, I have been able to reduce the summand to: \frac{1}{r^2-r+1} (\frac{1}{r} + \frac{1}{r+1}) and unfortunately I can't see a telescoping sum here :/ I'm still working on the problem though

Re: HSC 2013 3U Marathon Thread $Find$ \ \ \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{1}{3^{m+n}}

Re: HSC 2013 3U Marathon Thread Elegant, nice work Alternatively: 100^2-98^2 + 99^2-97^2 + 96^2-94^2 + 95^2-93^2 + \dots + 4^2-2^2 + 3^2-1^2 Difference of 2 squares (100-98)(100+98) + (99-97)(99+97) + \dots + (3-1)(3+1) Note that first bracket in each term is 2, take common...

Re: HSC 2013 3U Marathon Thread Note that the LHS is: \int_{0}^{1} x(1+x)^n \ dx (which we can derive easily) Use u-substitution, we yield the RHS.

Re: HSC 2013 4U Marathon In my opinion this is one of the few problems where induction isn't "cheap". Its not like we are getting the inequality out of nowhere since we can observe that its true for n=2 n=3 and n=4 rather easily. So its feasible to 'guess' the general case and prove by...

Re: HSC 2013 4U Marathon You are on the right track with geometric series. Make some strong manipulations so you can simplify your sum much more easily (or just grind out the result) That result is correct, it should be straightforward from there, I'm not sure how it would prove it for only...

Re: HSC 2013 3U Marathon Thread $Find$ \ \ \ 100^2+99^2-98^2-97^2+96^2+95^2-94^2-93^2+ \dots + 4^2 + 3^2 -2^2 -1^2 (The signs alternate in pairs) (Provide proof of your value, either with latex or a brief outline of steps taken)

I think you missed the point of his post entirely

Would directly emailing the board of studies work?

Ah yes I see what you mean by tessellate now, thank you that makes a lot of sense.