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So in this very lovely STEP question, I find the wording a little strange in part (ii). Is it asking for a proof or just for me to state the values? I asked a maths teacher and he is of the opinion that it is NOT asking for a proof. But the thing is, the next part is so easy after this part is completed, and therefore if it isn't asking for a proof then this is an extremely short STEP III question. I'm not sure what to make of it, I'd appreciate thoughts on what part ii means.
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(Original post by I hate maths)
So in this very lovely STEP question, I find the wording a little strange in part (ii). Is it asking for a proof or just for me to state the values? I asked a maths teacher and he is of the opinion that it is NOT asking for a proof. But the thing is, the next part is so easy after this part is completed, and therefore if it isn't asking for a proof then this is an extremely short STEP III question. I'm not sure what to make of it, I'd appreciate thoughts on what part ii means.
What year is it? Is there a mark scheme availible?
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the bear
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ii) is not a proof... just investigate the two possible values which arise.
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(Original post by the bear)
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(Original post by StayWoke)
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Mark scheme probably should have been first port of call for me as per StayWoke's sensible suggestion; I got sent this individual question and should've checked. Anyway the "mark scheme" (it doesn't show where marks are allocated) implies a proof. It basically says that by considering (F_nF_{n+3}-F_{n+1}F_{n+2})-(F_{n-2}F_{n+1}-F_{n-1}F_n) and applying the recurrence relation, the expression is zero and then it makes the conclusion. It doesn't explicitly say how many marks (if any) is given to that bit of algebra. I can imagine something like this being a little annoying in a high pressure exam where the wording could have been a little bit better. In fact probably quite a handful of students just stated the values as I might have done and possibly lost marks for no reason as the proof isn't too technical. This is STEP III 2012 if you were interested.
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Zacken
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(Original post by I hate maths)
Anyway the "mark scheme" (it doesn't show where marks are allocated) implies a proof. It basically says that by considering (F_nF_{n+3}-F_{n+1}F_{n+2})-(F_{n-2}F_{n+1}-F_{n-1}F_n) and applying the recurrence relation, the expression is zero and then it makes the conclusion. It doesn't explicitly say how many marks (if any) is given to that bit of algebra.
Here's the excerpt from the actual markscheme. FWIW, I think the question is quite clear in requiring a proof, but this is possibly with the experience of two years of uni maths and Cambridge Tripos exams.

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(Original post by Zacken)
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Thanks Zacken, this makes it crystal clear what the expectation is. I can imagine it being obvious to you as you say, but I, someone else on this thread, and even a maths teacher I asked were doubtful. It's just the word "find" rather than "show" or "prove". Perhaps the bloke who set this question thought the proof to be so trivial that "find" suffices if you know what I mean. And to be fair, the proof is not technical and I'd expect a high amount of A/A* further maths students to be able to complete it. Sorting out the semantics now has been very helpful though; I'll keep this in mind for future questions and I'll assume that if they ask me to find something, I'd need to provide a proof of my finding unless they explicitly say they don't want one.

As a side note where did you find this wonderful mark scheme?
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If you just look at (ii) in isolation, I agree that "find" makes it a little ambiguous. But part (i) strongly hints at what is going on in (ii), to the point that (ii) becomes trivial unless you're expected to actually prove your answer correct. And there's just no way that you're going to get (i) and (ii) of a 3 part STEP question being that trivial.
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(Original post by DFranklin)
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You raise a very good point, which I also made in my OP (that the question would be extremely short). Whenever I see a statement about natural numbers in these sorts of things, one of the things on my check list is induction anyway. Initially it does look like they're guiding you on some kind of proof by induction since part (i) looks like the basis, and part (ii) is the proof. And if it did say "prove" in part (ii) I would have just kept going like that straight away no problem.

In my humble opinion, I don't think a question should be difficult to actually understand the intention of the author from if possible (rather than the actual maths). The examiner's report for this question makes an interesting read: 83% of candidates attempted this question (most popular), and on average half marks were scored (third most well attempted). Part (i) "caused no problems" and part (iii) was "generally fairly well done", but part (ii) was "not well attempted, with a number stating the two values the expression can take but failing to do anything else or failing with the algebra".
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(Original post by I hate maths)
In my humble opinion, I don't think a question should be difficult to actually understand the intention of the author from if possible (rather than the actual maths).
I have a lot of sympathy for this point of view, particularly when it would have been easy to avoid any ambiguity by merely changing a few words.

On the other hand, a huge part of doing really well in these kinds of exams (and the Tripos, for that matter) is getting into the mind of the examiner and having a good idea of what is expected for each part of a question.

The examiner's report for this question makes an interesting read: 83% of candidates attempted this question (most popular), and on average half marks were scored (third most well attempted). Part (i) "caused no problems" and part (iii) was "generally fairly well done", but part (ii) was "not well attempted, with a number stating the two values the expression can take but failing to do anything else or failing with the algebra".
Again, on first glance that looks rather damning, but I'd say that part (ii) is also *significantly* harder than the other 2 parts.

And to be honest I'm very much doubting you had many people scoring well in their other question choices and at the same time thinking that writing "0 x 3 - 1 x 1 = -1 = 1 x 5 - 2 x 3" and "-1 when n is even, 1 when n is odd" was going to be sufficient to answer two parts of a 3 part STEP question.
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(Original post by DFranklin)
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You're completely right of course, I mean you sure know a hell of a lot more than me about anything that even smells like it has maths involved with it. FWIW I'll spoiler my solution to part (ii).

Spoiler:
Show


Basis n=0 true by part (i).

Assume that F_{k}F_{k+3}-F_{k+1}F_{k+2}=(-1)^{k+1}

RTP:  F_{k+1}F_{k+4}-F_{k+2}F_{k+3}=(-1)^k

Proof:

 LHS=F_{k+1}F_{k+4}-F_{k+2}F_{k+3} 



= F_{k+1}(F_{k+3}+F_{k+2})-(F_{k+1}+F_{k})F_{k+3}



= F_{k+1}F_{k+2}-F_{k}F_{k+3}



=-(-F_{k+1}F_{k+2}+F_{k}F_{k+3})



=(-1)^k



=RHS

By principle of mathematical induction bla bla bla true for n \in \mathbb{N}. Then some stuff about the different cases.

I wish I could write in LaTeX better and make that stuff come down in the middle of the page and add comments on the side without it going all janky but I can't. Anyway this is why I initially thought the question was strange, seeing as the proof was this easy (and recurrence relation induction is bog standard FP1). It was a combination of that and the word "find". But yeah, silly to assume they'll let a mere stating of the result slide so you're still right of course.

EDIT: I talk too much about maths and probably shouldn't have wasted your time on this haha. Thanks though I feel like I've gained stuff from this thread.
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