A Level maths sequences question

I understand how to prove b) without using calculus but the mark scheme says that using differentiation can only achieve 2 marks max even with explanation relating to discrete continuous functions or similar. While it may not be the best or quickest method, it seems mathematically valid to me so does anyone know why it wouldn't achieve full marks?

Mark Scheme (Page 7):

https://www.ocr.org.uk/Images/677010-mark-scheme-pure-mathematics-and-comprehension.pdf

Examiners report (Page 6):

https://www.ocr.org.uk/Images/687926-examiners-report-pure-mathematics-and-comprehension.pdf

Reply 1

mqb2766

Original post by 0-)

I understand how to prove b) without using calculus but the mark scheme says that using differentiation can only achieve 2 marks max even with explanation relating to discrete continuous functions or similar. While it may not be the best or quickest method, it seems mathematically valid to me so does anyone know why it wouldn't achieve full marks?
Mark Scheme (Page 7):
https://www.ocr.org.uk/Images/677010-mark-scheme-pure-mathematics-and-comprehension.pdf
Examiners report (Page 6):
https://www.ocr.org.uk/Images/687926-examiners-report-pure-mathematics-and-comprehension.pdf

The sequence is only defined for discrete values of n (domain) so you cant take limits as h->0 etc as occurs in calculus.

Reply 2

DFranklin

Original post by mqb2766

The sequence is only defined for discrete values of n (domain) so you cant take limits as h->0 etc as occurs in calculus.

So define

f(x) = \dfrac{x}{x+1}

; f(x) is defined for real x > 0 and f'(x) = 1/(x+1)^2 > 0, so (from my vague understanding of what's "sufficient justification" at A-level), f(x) is increasing. So

m > n \implies f(m) > f(n)

, and this is true whether or not you restrict m, n to be (+ve) integers.

For sure, if you argued "by the mean value theorem,

f(m)-f(n) = f'(\xi) (m-n)

for some

\xi \in (m, n)

, and since the RHS is always > 0 for m > n, f is strictly increasing", I'd be hard put to deduct marks at university,let alone A-level, but by the looks of things, this would still not score full marks.

Now, it's possible that people going the calculus route left logical holes, but If we're being honest (and considering the relatively low bar for rigour normally set at A-level), this seems like a clear case of "we didn't want to give marks for people using calculus", in which case I think the correct path would have been to specify this in the question.

Reply 3

0-)

Original post by mqb2766

The sequence is only defined for discrete values of n (domain) so you cant take limits as h->0 etc as occurs in calculus.

I get that but if you prove that the continuous function f(n)=n/(n+1) defined for all n>0 is increasing using differentiation, can’t you use that to prove that the discrete sequence is increasing since N is a subset of the positive reals?

(edited 11 months ago)

Reply 4

mqb2766

Original post by 0-)

I get that but if you prove that the continuous function f(n)=n/(n+1) defined for all R is increasing using differentiation, can’t you use that to prove that the discrete sequence is increasing since N is a subset of R?

See Dfranklins post above. Its possible with all the justification, but its not really what they wanted and for 3 marks the justification would be overkill.

Reply 5

DFranklin

Original post by 0-)

See my reply above (basically agreeing with you).

At the same time, I'd guess that the number of people who can correctly argue this at A-level (for example, what you said is wrong because f(n) isn't defined for all R as it's undefined when n = -1) who wouldn't just prove it algebraically is fairly small. But if you go that route I think it's also clear you're suddenly going "you need to be rigourous at A-level" when typically little to no rigour is expected. (For that matter, in a rigourous approach, you're not getting away with answering "the limit is 1" to "find the limit of a_n" without epsilon-delta justification, which they're obviously not expecting).

Reply 6

0-)

Original post by DFranklin

See my reply above (basically agreeing with you).
At the same time, I'd guess that the number of people who can correctly argue this at A-level (for example, what you said is wrong because f(n) isn't defined for all R as it's undefined when n = -1) who wouldn't just prove it algebraically is fairly small. But if you go that route I think it's also clear you're suddenly going "you need to be rigourous at A-level" when typically little to no rigour is expected. (For that matter, in a rigourous approach, you're not getting away with answering "the limit is 1" to "find the limit of a_n" without epsilon-delta justification, which they're obviously not expecting).

Yes I realised my mistake and edited my post soon after.

Reply 7

0-)

Original post by DFranklin

See my reply above (basically agreeing with you).
At the same time, I'd guess that the number of people who can correctly argue this at A-level (for example, what you said is wrong because f(n) isn't defined for all R as it's undefined when n = -1) who wouldn't just prove it algebraically is fairly small. But if you go that route I think it's also clear you're suddenly going "you need to be rigourous at A-level" when typically little to no rigour is expected. (For that matter, in a rigourous approach, you're not getting away with answering "the limit is 1" to "find the limit of a_n" without epsilon-delta justification, which they're obviously not expecting).

Yes I think what I found odd about the mark scheme for this question was that it seemed to require more rigour than normal for A Level. I’ve shown it to other people and they agreed with you that they should have explicitly told people not to use calculus.

Reply 8

DFranklin

Original post by 0-)

Yes I realised my mistake and edited my post soon after.

Sure - I wasn't meaning to be snarky (I don't consider that a "meaningful" mistake, more an oversight), But trying to point out that if the examiners suddenly go "justify everything", it's very easy to slip up.

[to be clear, "aesthetically" i don't think you should answer this using calculus. but if i didn't want someone to answer using calculus, i'd explicitly say so, I wouldn't just make the mark scheme say "if they use calculus, use every excuse you can think of to dock them marks"...]

(edited 11 months ago)

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