# How would you approach and solve these maths problems?

So, I am doing some mathematical problems to prepare for a university entrance exam tomorrow and I am confused how to approach and solve these questions quickly and efficiently.

I had checked the mark scheme to these questions but I don’t quite understand the method they stated.
(edited 11 months ago)
Original post by mellow2006
So, I am doing some mathematical problems to prepare for a university entrance exam tomorrow and I am confused how to approach and solve these questions quickly and efficiently.

I had checked the mark scheme to these questions but I don’t quite understand the method they stated.

For 12) Id simply note there is going to be a lot of cancelling due to -1 and +1, so Id write out the first few terms (and last terms) and spot the pattern. Though a bit more insight would give that all the terms will appear 3 times so +3 or -3, apart from the firt couple and the last couple.

15) just uses the simple difference where you spot what the x is, so for a) its [x] + [x] so a simple method of differences/telescoping for a small series. A should be clear, as is B then it generalise to C and D stands out like a sore thumb

16) Each numerator is twice the previous denominator so a lot of cancelling occuirs
(edited 11 months ago)
Original post by mqb2766
For 12) Id simply note there is going to be a lot of cancelling due to -1 and +1, so Id write out the first few terms (and last terms) and spot the pattern. Though a bit more insight would give that all the terms will appear 3 times so +3 or -3, apart from the firt couple and the last couple.

Not sure what are you referring to when you say "all the terms appear 3 times so +3 or -3". Surely each summand is not +3 or -3.

My hint would be to notice that first two terms or equivalently last two terms always cancel (odd power and even power of -1), so you are left with the sum of (-1)^n or (-1)^(n+2) equivalently, whichever you like more.
Original post by hassassin04
Not sure what are you referring to when you say "all the terms appear 3 times so +3 or -3". Surely each summand is not +3 or -3.

My hint would be to notice that first two terms or equivalently last two terms always cancel (odd power and even power of -1), so you are left with the sum of (-1)^n or (-1)^(n+2) equivalently, whichever you like more.

Its similar, just depends on how you group the terms. (-1)^0 occurs once, (-1)^1 occurs twice, (-1)^3 occurs three times as do most of the subsequent terms apart from the last two.... Its one way to group/count things.
(edited 11 months ago)
Original post by mqb2766
Its similar, just depends on how you group the terms. (-1)^0 occurs once, (-1)^1 occurs twice, (-1)^3 occurs three times as do most of the subsequent terms apart from the last two.... Its one way to group/count things.

So clearly not all terms ( n-th powers of -1 ) occur three times as you yourself mentioned. Your way sounds unnecessarily more complicated to me but if some find it more natural/obvious then sure.
Original post by hassassin04
So clearly not all terms ( n-th powers of -1 ) occur three times as you yourself mentioned. Your way sounds unnecessarily more complicated to me but if some find it more natural/obvious then sure.

Probably fair, but there are several similar ways to view it and its just a case of getting something that works. Another similar way is just to note (-1)^0 = 1 and all subsequent terms can be paired/cancel.
(edited 11 months ago)
Original post by mqb2766
For 12) Id simply note there is going to be a lot of cancelling due to -1 and +1, so Id write out the first few terms (and last terms) and spot the pattern. Though a bit more insight would give that all the terms will appear 3 times so +3 or -3, apart from the firt couple and the last couple.

15) just uses the simple difference where you spot what the x is, so for a) its [x] + [x] so a simple method of differences/telescoping for a small series. A should be clear, as is B then it generalise to C and D stands out like a sore thumb

16) Each numerator is twice the previous denominator so a lot of cancelling occuirs

I know this is way past when the OP will care (I don't look at f46 that often), but given other discussions, thought it would be worth adding my thoughts on how I'd answer these in an exam:

12) I'd immediately note two 'sub terms' in each term cancel, so $x_n = (-1)^n$. (not a big deal).
15) I'll admit to wasting a little time actually looking at differences etc., but it's pointless here. Once you actually look at D and E, it's obvious both can't be true, and then E is obviously "small", while the LHS is > 1/2. So E is false.
16) Just looking, taking a 2 from each term in the top means a lot cancels, the answer's going to be of the form 2^n p/q for some choice of integers p, q.
When n = 2, the answer is 2. That rules out 3|q and means only one possible answer.

The general point here is that of the 3 questions, in two of them I'm not really solving the problem, I'm somewhat "gaming the system" to rule out options until I can determine which of A-E to answer. 16 is a fairly common example where I can use mathematical intuition to say "the answer's going to look roughly like XYZ" and then use the given answers and particular cases to work out the correct one.

The "general general" point: I find I'm naturally quite good at playing these games on the MAT questions. But I felt it ended up counterproductive for the SMC. How much that's my particular mathematical strengths / weaknesses (geometry!) and how much it's a genuine difference in "question setting" I don't know (and I do feel the MAT is a bit too "gameable"). But I definitely found they didn't "play the same" for me.
Original post by DFranklin
I know this is way past when the OP will care (I don't look at f46 that often), but given other discussions, thought it would be worth adding my thoughts on how I'd answer these in an exam:

12) I'd immediately note two 'sub terms' in each term cancel, so $x_n = (-1)^n$. (not a big deal).
15) I'll admit to wasting a little time actually looking at differences etc., but it's pointless here. Once you actually look at D and E, it's obvious both can't be true, and then E is obviously "small", while the LHS is > 1/2. So E is false.
16) Just looking, taking a 2 from each term in the top means a lot cancels, the answer's going to be of the form 2^n p/q for some choice of integers p, q.
When n = 2, the answer is 2. That rules out 3|q and means only one possible answer.

The general point here is that of the 3 questions, in two of them I'm not really solving the problem, I'm somewhat "gaming the system" to rule out options until I can determine which of A-E to answer. 16 is a fairly common example where I can use mathematical intuition to say "the answer's going to look roughly like XYZ" and then use the given answers and particular cases to work out the correct one.

The "general general" point: I find I'm naturally quite good at playing these games on the MAT questions. But I felt it ended up counterproductive for the SMC. How much that's my particular mathematical strengths / weaknesses (geometry!) and how much it's a genuine difference in "question setting" I don't know (and I do feel the MAT is a bit too "gameable"). But I definitely found they didn't "play the same" for me.

I think the questions are tmua but agree with the general points youre making above. For 15) the "answers" pretty much lead you through the question and as you say it has to be either D or E as false. The A option is very well known and B and C are then trivial simple examples of it. Somewhat similar for 18). Both the form of the multi choice answers and the simple cancelling that goes on in both series means that ithey could asked pre a level.

Ive not been through the mat this year, but heard that the multichoice was a bit harder than usual. Ill do that later. Ive generally found that the smc stuff is somewhat useful for it, though somewhat is obviously vague.. In both cases, there are quite a few quesitons where the multichoice answers can be used to as a hint / guide to work out what the answer actually is.
Original post by mqb2766
I think the questions are tmua but agree with the general points youre making above. For 15) the "answers" pretty much lead you through the question and as you say it has to be either D or E as false. The A option is very well known and B and C are then trivial simple examples of it. Somewhat similar for 18). Both the form of the multi choice answers and the simple cancelling that goes on in both series means that ithey could asked pre a level.
I'm sure you're right about the TMUA - the MAT questions do vary in difficulty (each paper they start easy and get harder) and I thought this was maybe a compilation of more "approachable ones". The feel is quite similar though, I think.

Q15 is actually a bit annoying - as you said, it's clearly been written to "lead you through" the correct mathematical approach, but that's basically irrelevant given you have to find the one false statement.

Q18) - the "TMUA" comment is perhaps relevant (since it's not remotely hard to do it properly), but I'd say it's fairly common to see "late" MAT questions where it's not hard to see something like "we have N steps, and in steps k= 2, ..., N-1 we add log k, so the answer's going to have log(N!) in it somewhere" and then rather than worry in detail about the end points you can reason from the answers that have log(N!) in them and a small concrete example.

Ive not been through the mat this year, but heard that the multichoice was a bit harder than usual. Ill do that later. Ive generally found that the smc stuff is somewhat useful for it, though somewhat is obviously vague.. In both cases, there are quite a few quesitons where the multichoice answers can be used to as a hint / guide to work out what the answer actually is.
I saw the "10 minute rush through" video Oxford put out and did think they looked a little harder than usual (although it was also 1am and I'd been drinking, so, um...)

I wouldn't dispute the SMC is "somewhat" useful, although I suspect if it was the "main prep outside A-level" you might be a bit slow to spot 'gameable' questions (my feeling was "process of elimination without actually solving the question" tended to be quite difficult for most SMC questions).
Original post by DFranklin
I wouldn't dispute the SMC is "somewhat" useful, although I suspect if it was the "main prep outside A-level" you might be a bit slow to spot 'gameable' questions (my feeling was "process of elimination without actually solving the question" tended to be quite difficult for most SMC questions).

There are actually a reasonable number of smcs you can do like that or use the answers to narrow things down. There was a classic one a couple of years ago (imc) where the hard (Q25) geometry question had the answers given as surds. From the diagram the answer was clearly a smidgen under 1, and a rough approximation of the surds gave just one answer satisfied that. Id probably say that the problem solving stuff (and checking answers/working backwards is part of that) is probably the main thing to transfer from smc to mat/tmua.

Edit - the question I was thinking of was
https://ukmt.org.uk/wp-content/uploads/2023/08/imc-2017-q.pdf
The radius is 2 so x is a bit under 1 as 2-x must be a bit greater than 1 for the areas to be equal. Some of the answers are "stupid" so D is > 2 for instance. A is the only sensible value. There are quite a few questions in a similar vein, and the smc is pretty much the same.
(edited 10 months ago)