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Re: HSC 2013 4U Marathon I got, \frac{13\sqrt{2}}{6} - \frac{7}{3} = 0.73079.... I want to figure out the bonus before I post my solution for this one.

Re: HSC 2013 4U Marathon I'm afraid you would have to prove the sum converges in order to find it, however I doubt that the Sydney grammar teachers who made this question expected students to do so. u_{n+1} = u_n + u_n^2 \ \Rightarrow \ \frac{u_{n}}{u_{n+1}} = \frac{1}{1+u_n}...

Re: HSC 2013 3U Marathon Thread Yep that worked for me Well done ======== $Prove for integers$ \ n \ $and$ \ k \ \ n>2k \binom{n}{k} 2^k = \binom{n}{n} \binom{n}{k} + \binom{n}{n-1} \binom{n-1}{k-1} + \binom{n}{n-2} \binom{n-2}{k-2} + \dots + \binom{n}{n-k} \binom{n-k}{0}

Re: HSC 2013 3U Marathon Thread Sorry you are right braintic, I am missing a m/1!, I apologise, fixing now.

Re: HSC 2013 3U Marathon Thread I got that question from Summation of Series - Jolley, series number 196. I'm pretty sure the 1 cancels out which is why it's there EDIT: Let me redo it I may have made a mistake hahahahaha

Re: HSC 2013 2U Marathon Yep that is correct well done. $Define a function$ \ y=F(x) \ \ $such that for some value$ \ x, \ \ F(x) \ $is the greatest integer less than or equal to $ \ x $For example$ \ F(1.5) = 1 \ \ F(\pi) = 3 \ \ F(4.8) = 4 \ \ F(5) = 5 $i) Sketch the graph of$ \ y=F(x)...

Re: HSC 2013 4U Marathon Sure I'll just leave all questions here that are old to be unanswered rather than post the solutions for other people to benefit No need to be sarcastic about it =)

Re: HSC 2013 4U Marathon Because some people want to know solutions of these questions? In the past people have asked for solutions to a lot of these questions, and I thought this thread would be of little benefit to people if they don't know the solutions to these questions.

Re: HSC 2013 4U Marathon First transforming the wanted inequality so its easier to solve: n^n \geq (n+1)^{n-1} \left ( \frac{n}{n+1} \right )^n \geq \frac{1}{n+1} \ \ \ (*) This is what we need to prove, transforming the above AM-GM inequality: \left( \frac{a_1 + a_2 + \dots + a_n}{n}...

Re: HSC 2013 2U Marathon $i) Find real numbers A and B, such that$ \ \ \frac{A}{x} + \frac{B}{x+1} = \frac{1}{x(x+1)} $ii) Hence find$ \ \int \frac{dx}{x(x+1)} $iii) Show by expanding the sum out$ \sum_{k=1}^N \left(\frac{1}{k} - \frac{1}{k+1} \right ) = 1 - \frac{1}{N+1} $iv) Hence...

Re: HSC 2013 2U Marathon Yep well done, however for the last part I would have liked some better justification of how the term, (n-1)x^{n+1} converges to zero One way is that: \lim_{n \to \infty} (n-1)x^{n+1} = \lim_{n \to \infty} nx^{n+1} - \lim_{n \to \infty} x^{n+1} We know the second...

Re: HSC 2013 2U Marathon $i) Explain why$ \ 1 + 2 + 3 + \dots + n = \frac{n}{2} (n+1) $ii) Show that$ \ (k+1)^3 - k^3 = 3k^2 + 3k + 1 $iii) By taking the sum of both sides from$ \ k=1 \ $to$ \ n $ of the result in part (ii), and expanding the sum out, show that$ 1^2+ 2^2+ 3^2 + \dots +...

Re: HSC 2013 4U Marathon $Prove for some integers$ \ m, n \ m>n \binom{m}{0} - \binom{m}{1} + \binom{m}{2} - \dots + (-1)^n \binom{m}{n} = (-1)^n \binom{m-1}{n} (above results and the one in the 3U thread are verified since they come from the book, Summation of Series - Jolley)

Re: HSC 2013 3U Marathon Thread $Prove for some integers$ \ m, n 1+ \frac{m}{1!} + \frac{m(m+1)}{2!} + \dots + \frac{m(m+1) \dots ( m+n-1)}{n!} = \frac{(m+n)!}{m! n!}

Re: HSC 2013 4U Marathon $Prove for some integer$ \ n \binom{n}{1} - \frac{1}{2} \binom{n}{2} + \frac{1}{3} \binom{n}{3} - \dots + \frac{(-1)^{n-1}}{n} \binom{n}{n} = \sum_{k=1}^n \frac{1}{k}

Re: HSC 2013 3U Marathon Thread o_o This year or last year?

Re: HSC 2013 3U Marathon Thread $2 projectiles are fired from the origin at the same time with the same velocity with angles of projection$ \ \alpha \ $and$ \ \beta \frac{\pi}{2} > \alpha > \beta > 0 $Prove that if the projectiles collide then the horizontal distance from the origin is...

Re: HSC 2013 3U Marathon Thread Yep I redid and you are right if we take range as distance from projection (which is probably the more correct definition), I took range as distance from origin.

Re: HSC 2013 2U Marathon Yep thanks, changing it now

Re: HSC 2013 2U Marathon $i) Explain why$ \ \ 1+ x + x^2 + x^3 + \dots + x^{n-1} = \frac{x^{n} - 1}{x-1} $ii) By differentiating, prove that$ x + 2x^2+3x^3+ \dots + (n-1)x^{n-1} = \frac{(n-1)x^{n+1} - nx^{n} + x}{(x-1)^2} $iii) You may assume$ \ \lim_{n \to \infty} nx^n = 0 \ $if$ \...