ATAR Notes: Forum
HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Extension 1 => Topic started by: joengelman on April 15, 2020, 11:02:41 am
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Hey guys,
Just curious, when doing the proof questions on the MX1 topic test, I've noticed that a lot of the series questions are just proving one thing? Like at my school, we've learnt to do questions in series, like 1+4+9+n=2n+1, not like prove x^k=(1-x^n)/1-x, where there are like two variables (being n and k within the same equations). Have I missed something? I mean, Maths in focus + New senior doesn't have those sorts of questions, but maybe I'm looking at the wrong textbooks?
Thanks,
Jo
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It's a sum.
\[ \sum_{k=1}^{n-1} x^k\text{ is a shorthand for }1+x+x^2+\dots+x^{n-1}.\\ \text{So in that question, what you're trying to prove by mathematical induction is the statement}\\ \boxed{ 1+x+x^2+\cdots+x^{n-1} = \frac{x^n-1}{x-1}}. \]
Which happens to be the sum of terms in a geometric sequence. In particular, with first term \(1\) and common ratio \(x\).
Sigma notation is actually no longer a part of Adv/MX1 though - it's exclusively an MX2 concept. I told the author that just now so hopefully that issue will get addressed. For now though, I'd suggest just proving that equivalent statement I boxed instead.
(Note however that NESA allows use of sigma notation in MX1 as a shorthand convenience. It's just not taught explicitly.)