# AQA A-Level Maths 2021 Paper 1 Question (13c)

How did you all find this question? I feel I wasn’t particularly well prepared to answer this one. It seems a bit out of syllabus. They use the fact that three consecutive numbers must include an even number and a multiple of three. It’s sort of intuitive when you think about but there’s no way someone could think of that during the test. It’s also not covered in the OUP book. Was it covered in whatever books you’re using or in your college?
Original post by HaziqALevel
How did you all find this question? I feel I wasn’t particularly well prepared to answer this one. It seems a bit out of syllabus. They use the fact that three consecutive numbers must include an even number and a multiple of three. It’s sort of intuitive when you think about but there’s no way someone could think of that during the test. It’s also not covered in the OUP book. Was it covered in whatever books you’re using or in your college?

Not sure if we're looking at the same question:
"Hence, prove that 250n3 þ 300n2 þ 110n þ 12 is a multiple of 12 when n is a positive whole number"
https://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2021/november/AQA-73571-QP-NOV21.PDF

If we are, then the question is testing you on using proofs alongside using sequences. If you have done enough proof questions, this should be OK-ish to do.
Original post by MindMax2000
Not sure if we're looking at the same question:
"Hence, prove that 250n3 þ 300n2 þ 110n þ 12 is a multiple of 12 when n is a positive whole number"
https://filestore.aqa.org.uk/sample-papers-and-mark-schemes/2021/november/AQA-73571-QP-NOV21.PDF

If we are, then the question is testing you on using proofs alongside using sequences. If you have done enough proof questions, this should be OK-ish to do.

Yeah that’s the one. I think knowing that three consecutive numbers must include an even number and a multiple of three is essential to be able to solve this one. They just quote this fact without any justification in the mark scheme and in hindsight it seems kind of obvious. I’ve done a fair amount of proof questions but haven’t come across one of this type. Maybe I’m not looking in the right place. Were you able to solve this one?

I actually remember noticing that for each value of n one of the terms was even and another was a multiple of 3 but I wasn’t able to justify it formally. Perhaps if I had just stated it as fact, without worrying about justifying it, and used it to complete the proof I would’ve still got the marks?
Original post by Haziq123
Yeah that’s the one. I think knowing that three consecutive numbers must include an even number and a multiple of three is essential to be able to solve this one. They just quote this fact without any justification in the mark scheme and in hindsight it seems kind of obvious. I’ve done a fair amount of proof questions but haven’t come across one of this type. Maybe I’m not looking in the right place. Were you able to solve this one?

I actually remember noticing that for each value of n one of the terms was even and another was a multiple of 3 but I wasn’t able to justify it formally. Perhaps if I had just stated it as fact, without worrying about justifying it, and used it to complete the proof I would’ve still got the marks?

I will have to work out 13b in order to understand what is required of 13c
P(x) = 125x^3 +150x^2 + 55x + 6
P(x) = (5x+1)(25x^2 + 25x + 6)
P(x)=(5x+1)(5x+2)(5x+3)

13(c):
"250n^3 + 300n^2 + 110n + 12 = 2(5n+1)(5n+2)(5n+3)
Let 5n = a
2(a+1)(a+2)(a+3)
...[Run through a series of numbers from say 0 to 5 showing the effects.]...
Because of the difference between the factors, there will always be a set of factors where you have the following 2*2*3 [you can alternatively say there will always be a multiple of 2 and a multiple of 3, then the factor of 2], which makes the resulting number always divisible by 12. Therefore...."

I think this comes under proof by deduction.
You would otherwise need to know that (x+1)(x+2)(x+3) will always result in a multiple of 6.

I am not entirely sure whether the examiner would be willing to offer full marks for the above, but it's how I would approach the question.
(edited 1 year ago)
Original post by HaziqALevel
How did you all find this question? I feel I wasn’t particularly well prepared to answer this one. It seems a bit out of syllabus. They use the fact that three consecutive numbers must include an even number and a multiple of three. It’s sort of intuitive when you think about but there’s no way someone could think of that during the test. It’s also not covered in the OUP book. Was it covered in whatever books you’re using or in your college?

I've seen a fair amount of Year 1 proof questions that are structured in this way. Every time you come across a question that says "prove f(x) is a multiple of something" just list all the common factors of that number. For example if it was 6, you could prove one to be a factor of 2 and the other to be a factor of 3, hence when you multiply them together you get a multiple of 6.

It just requires you in some cases to think outside of the box, and ALWAYS pay attention to consecutive terms, as this is a dead giveaway on how to go about answering these types of questions.