HSC 2013-14 MX1 Marathon (archive) (6 Viewers)

FrankXie · Nov 24, 2014

Re: HSC 2013 3U Marathon Thread

Sy123 said:
$\\ $EXTENSION (still within 3U knowledge), can you find another non-inductive method of finding$ \ a_n \ $(that is derivable without knowing the result as well (i.e. not ad-hoc))?$$

Nice one.

$Consider $ b_n=\frac{1}{a_n}. $ flip the recursive relationship, it is clear that $ b_n $ is an AP.$

FrankXie · Nov 24, 2014

Re: HSC 2013 3U Marathon Thread

Nukeboy said:
View attachment 30931

Harder 2unit question

Let the point be P. From similar triangles, P divides internally the interval AB in the ratio 3:2.

FrankXie · Nov 24, 2014

Re: HSC 2013 3U Marathon Thread

dunjaaa said:
View attachment 30876 Harder 2u

$29+12\sqrt5=29+2\sqrt{180}=(\sqrt{20})^2+(\sqrt9)^2+2\sqrt{20}\times\sqrt9=(\sqrt{20}+\sqrt9)^2$

$\therefore \sqrt{29+12\sqrt5}=\sqrt9+\sqrt{20}=3+2\sqrt5$

$and $ 11+6(3+2\sqrt{5})=29+12\sqrt5 $ again$

$\therefore \sqrt{11+6\sqrt{29+12\sqrt5}}=3+2\sqrt5$

$$ Finally $ \sqrt{53+12(3+2\sqrt5)}=\sqrt{89+24\sqrt5}=\sqrt{89+2\sqrt{720}}$

$=\sqrt{80+9+2\sqrt{80\times9}}=\sqrt{(\sqrt{80} + \sqrt{9})^2}=3+4\sqrt5$

Sy123 · Nov 29, 2014

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} \\ \\ $ii) Hence using the Binomial theorem, show that$ \\ \\ \binom{k}{k} + \binom{k+1}{k} + \dots + \binom{n}{k} = \binom{n+1}{k+1}$

HSC 2013-14 MX1 Marathon (archive) (6 Viewers)

FrankXie

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FrankXie

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Sy123

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