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I only really like and appreciate analytic-ish philosophy. I say -ish because analytic philosophy was a term that refers only to philosophy in the english world since the 20th century, however that type of rigorous systematic philosophy had been practiced in the medieval scholastic era and also...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Let$ \ T_N = \sum_{k=1}^N \frac{1}{\sqrt{k+\frac{1}{2}}} \\ S_N = \sum_{k=1}^N \frac{1}{\sqrt{k}} \\ \\ $Find$ \ \lim_{N \to \infty} \frac{T_N}{S_N}

Re: HSC 2015 4U Marathon \\ $Solve the following system of equations$ \\ \\ 2x_1 + x_2+x_3 + x_4 + x_5 = 6 \\ x_1 + 2x_2 + x_3+x_4+x_5 = 12 \\ x_1+x_2+2x_3+x_4+x_5 =24 \\ x_1+x_2+x_3+2x_4+x_5 = 48 \\ x_1+x_2+x_3+x_4+2x_5=96

Re: HSC 2015 4U Marathon \\ $Show that$ \ \log_{10}8 \ $is irrational$

Re: HSC 2015 3U Marathon (If you allow me to build off of this) \\ $ii) Using a graph, show that$ \\ \int_{n}^{2n}\frac{dx}{x} - \frac{1}{2n} < \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} < \int_{n}^{2n} \frac{dx}{x} \\ $iii) Hence explain why$ \ \lim_{n \to \infty}...

Your error occured when you integrated, there shouldn't be the \frac{-1}{n-1} If: \\ u = (1+x^2)^{-n} \\ du = -2nx(1+x^2)^{-n-1} \ dx \\ dv = 1 \ dx \\ v = x So the first term is \frac{x}{(1+x^2)^{n}} \right |_0^1

Re: HSC 2015 4U Marathon - Advanced Level Isn't the nth partial sum of the sequence S(n) = b1 + b2 + .. + bn? Therefore, S(1) = b1?

Re: HSC 2015 3U Marathon \\ $Using the identity$ \ (1+x)^n \left(1+ \frac{1}{x} \right)^n = \frac{1}{x^n}(1+x)^{2n} \ $or otherwise, show that$ \\ \\ \sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}

Re: HSC 2015 4U Marathon - Advanced Level \\ $The definitions of the sequences are$ \ a_{n} + a_{n+1} = b_n \ (1) \ $and$ \ a_n a_{n+1} = 2^n \ (2) \\ a_1 = 1 \ $and$ \ b_1 = 3 \\ \frac{2^{n+1}}{2^n} = \frac{a_{n+2} a_{n+1}}{a_{n+1}a_n} \Rightarrow \ a_{n+2} = 2a_n \\ $So, given$ \...

Re: HSC 2015 4U Marathon - Advanced Level \\ $By drawing the graph of$ \ f(x) = \tan^{-1} \left( \frac{1}{x} \right) \ $and drawing the upper rectangles from$ \ x = 1 \ $to$ \ x = N \ $for a positive integer$ \ N \\ $It becomes clear that$ \\ \\ \int_1^{N+1}\tan^{-1} \frac{1}{x} \ dx <...

Re: HSC 2015 4U Marathon - Advanced Level \\ \frac{A_0}{x} + \frac{A_1}{x+1} + \dots + \frac{A_n}{x+n} = \frac{1}{x(x+1) \dots (x+n)} \\ \\ \Rightarrow \ \sum_{k=0}^n A_k \frac{x(x+1)\dots(x+n)}{x+k} = 1 \\ x = -k \Rightarrow \ A_k (n-k)(n-k-1)\dots(1)(-1)(-2)\dots(-k) = (-1)^k(n-k)! k! \\...

Re: HSC 2015 3U Marathon Well give it a go if you think it will work. Also for that expression, I don't know how you simplified: \binom{n-1}{k} = \frac{(k+n)(k-1+n)}{n!} because that is not true

Re: HSC 2015 3U Marathon How would you simplify the LHS?

Re: HSC 2015 4U Marathon - Advanced Level f(xy) = f(x) f(y) - f(x+y) + 1 \\ $Let$ \ x=y=0 \ \Rightarrow \ f(0) = f(0)^2 - f(0) + 1 \Rightarrow \ f(0) = 1 \\ $Let$ \ x = 1, y = -1 \ \Rightarrow \ f(-1)(f(1) - 1) = 0 \\ \\ \therefore \ f(-1) = 0 \ $or$ \ f(1) = 1 \\ $If$ \ f(1) = 1 \ $then...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Show that$ \ \left (\frac{n+1}{2} \right)^n \geq n! \ $for integers$ \ n \geq 1

Re: HSC 2015 3U Marathon \\ $Show that$ \ \binom{k}{k} + \binom{k+1}{k} + \dots + \binom{n-1}{k} = \binom{n}{k+1} \ $for$ \ 0 \leq k \leq (n-1) \ $for integers$ \ k,n

Re: HSC 2015 4U Marathon - Advanced Level (Doesn't fit in the other thread) \\ $Show that there does not exist a polynomial$ \ f(x) \ $such that$ \\ xf(x-1) = (x+1)f(x)

Re: HSC 2015 3U Marathon This would require proof

Re: HSC 2015 3U Marathon \\ $Prove using first principles that$ \ \frac{d}{dx} \sin x = \cos x

Re: HSC 2015 4U Marathon \\ $Show that the hyperbola$ \ x^2 - y^2 = a^2 \ $and$ \ xy = c^2 \ $intersect at right angles$