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Re: HSC 2014 4U Marathon \\ $Let$ \ F_n = \frac{1}{1+ \frac{1}{1 + \frac{1}{\dots}}} \ $where there are$ \ n \ $fractions$ \\ \\ $So, $ F_1 = \frac{1}{1} \ , \ F_2 = \frac{1}{1+ \frac{1}{1}} = \frac{1}{2} \ $and so on$ $i) Show that$ \ F_{n+1} = \frac{1}{1+ F_n} \ $for$ \ n > 1 $ii) Prove...

yeah I'm not going to waste my post: ----------------------------------- There is no meaning of life on atheism, I think this is clear to anyone who really considers the question properly. If you say, "the meaning of life is happiness" or "the meaning of life is to stay alive and...

\\ \omega = \cos \left(\frac{2\pi}{9}\right) + i\sin \left(\frac{2\pi}{9}\right) \\ \\ $i) Due to De Moivere's theorem: $ \\ \omega^k = \left(\cos \left(\frac{2\pi}{9}\right) + i\sin \left(\frac{2\pi}{9}\right) \right)^k = \cos \left(\frac{2\pi k}{9} \right) + i\sin \left(\frac{2\pi k}{9}...

Re: HSC 2014 4U Marathon - Advanced Level Yep that was my method when seanieg89 posted this question near the start of the year (yes I am guilty of recyling some questions)

Re: HSC 2014 4U Marathon - Advanced Level At this part of your reasoning, are you saying that since: 1 + m > 2 \Rightarrow \ 2 > \sqrt[n]{1+m} because that does not seem to be a valid inference

Re: HSC 2014 4U Marathon - Advanced Level :(

Re: HSC 2014 4U Marathon - Advanced Level \\ $Prove for integers$ \ m,n > 1 \ $that$ \ (1+m)^{-\frac{1}{n}} + (1+n)^{- \frac{1}{m}} > 1

Re: HSC 2014 4U Marathon - Advanced Level So do you want a finite sum or infinite sum?

Re: HSC 2014 4U Marathon - Advanced Level Right. My bad.

Re: HSC 2013 3U Marathon Thread NOTE: Definitely HSC worthy, would expect this question to be in Q14 or even Q13 \\ $i) Prove using mathematical induction that, for some integer$ \ n \geq 0 \ $and real$ \ x \neq 0 \\ \\ 1 + (1+x) + (1+x)^2 + \dots + (1+x)^n = \frac{(1+x)^{n+1} - 1}{x} \\ \\...

Re: HSC 2014 4U Marathon - Advanced Level (HSC Methods only) \\ $Let$ \ H_n = \sum_{k=1}^n \frac{1}{k} \\ \\ $Show that$ \ \lim_{n \to \infty} (H_n - \ln n) \ $is finite$ --- \\ $If$ \ S_n = \sum_{k=1}^n \frac{(-1)^k}{k} \\ \\ $Show that$ \ S_{2n} = H_n - H_{2n} \\ \\ $Hence show that$ \ -...

Re: HSC 2014 4U Marathon Tell the mods to change the name, better to use the same thread so people have access to older questions Like in the other marathon

Poor bloke

Re: HSC 2014 4U Marathon Similar to what I have in mind, I just translate the problem into showing: $arg$ \left(\frac{z-1}{z+1} \right) = \frac{\pi}{2} \ $for$ \ |z| = 1 \\ \\ \frac{z-1}{z+1} = \frac{(z-1)(\bar{z} + 1)}{|z+1|^2} = \frac{|z|^2 - 1 + (z - \bar{z})}{K} \\ \\ = C(z - \bar{z}) \...

So your argument goes: 1 - If Islam or Christianity is true, then they ought to teach us morally good values 2 - Islam and Christianity teach bad moral values 3 - Therefore, Islam and Christianity are false --- The premise for dispute is premise number 2 (and as I will show, if...

No, it is literally true

Yeah some religious people definitely have bad and untenable beliefs, I agree, but that doesn't mean religion in general is bad or false. If you really think that 1st century people thought that heaven was literally in the sky, then I think you have been misled, people were not that ignorant at...

Did you read my post where I refuted your views? Did you comprehend it? Do you need me to explain it for you?

what caused the big bang? The Big Bang is an occurent event, thus it must have a cause, as all occurents have a cause

Your stardust is a contingent thing as it is given to motion and change and other occurents So where did you get your stardust from?