CfE Advanced Higher Mathematics 2017/2018

Matrices need to be abolished immediately

Needing a bit of help with a proof by induction question.

Prove that 8^n is a factor of (4n)! for all n >= 1

I'm attempting it now. Which part of induction are you on?

Are you meaning with regards to the course or that specific question?

The question.

If you're on part three (prove it works of n = k+1):
Notice that (4(k+1))! can be written as (4k+4)(4k+3)(4k+2)(4k+1)(4k)!, then notice that (4k+4) and (4k+2) can both be factorised.

So, two weeks ish to the exam and only starting studying now! Any tips/books/miracles anyone could send my way?

So, two weeks ish to the exam and only starting studying now! Any tips/books/miracles anyone could send my way?

There's a lot more than could be covered in 2 weeks here, but this is good if you need help with specific topics https://www.advancedhighermaths.co.uk/ (also has lots of old past papers and exam questions by topic, etc.)

I haven't studied at all beyond doing past papers in class, but I'll recommend watching these two series on linear algebra and calculus:
https://youtu.be/WUvTyaaNkzM?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr (calculus)
https://youtu.be/kjBOesZCoqc?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
(linear algebra)

The first series is entirely advanced higher/higher calculus and explains away a lot of the seemingly random formulas or rules you're told to memorise, which for me helps remembering them or if there is a question which you haven't been taught by rote.

If you've seen those and you still don't understand anything, or you're a big nerd, you could try watching calc 1/2 + the start of calc 3 lectures by this guy:
https://youtu.be/fYyARMqiaag?list=PLF797E961509B4EB5 (calc 1, a mix of higher/advanced higher)
https://youtu.be/H9eCT6f_Ftw?list=PLDesaqWTN6EQ2J4vgsN1HyBeRADEh4Cw- (calc 2, mostly advanced higher calculus)
https://youtu.be/tGVnBAHLApA?list=PLDesaqWTN6ESk16YRmzuJ8f6-rnuy0Ry7 (calc 3, start of this contains some lectures on the dot product, cross product, etc)

Although I'd recommend putting them on 2x speed or something because they are very long. It's much more in depth than the other videos.

There's also a guy on youtube that does solutions to past papers. DLBmaths iirc. Perhaps he'll be useful too.

Thanks guys, also have to sit the unit 3 nab in the next week. Will post back if I need any more advice( very likely).

So, two weeks ish to the exam and only starting studying now! Any tips/books/miracles anyone could send my way?

Struggling with questions that ask for a specific term in the binomial expansion, could someone helo and explain the method? thanks.

For $(a+b)^n$ where n is 2,3,4,... etc.
The binomial expansion is
$\sum_{r=0}^{n} {\binom{n}{r} a^{n-r}b^{r}}$
The "general term" is
$\binom{n}{r} a^{n-r}b^{r}$

When you're asked to find the coefficient of the term with a certain power, for example in this question (2016 paper):


the idea is to seperate each of the coefficients/x terms in the general term, e.g.:
$\binom{13}{r} {(\frac{3}{x})}^{13-r}{(-2x)}^{r}$
into
$\binom{13}{r} 3^{13-r} (-2)^r x^{-(13-r)} x^r$
$\binom{13}{r} 3^{13-r} (-2)^r x^{-(13-r)+r}$
$\binom{13}{r} 3^{13-r} (-2)^r x^{2r-13}$
It asks you for the coefficient of the term with x^9 in it, so 2r-13 = 9. [If it asks for "the term independent of x" it just means the term with x^0] Solve that and you get r = 11. Now since you know what r is for x^9, you can just sub it into the equation above and you get
$\binom{13}{11} 3^{13-11} (-2)^{11}=-1437696$

Hope this helps

Wow that was really helpful thanks.

I'm guessing you're beyond advanced higher level?

So you split it up and have the numbers on one side and the Xs on the other. Is it because they are both singular Xs that you can put the x^r with the other x^13-r ??

I'm still at advanced higher I just went out of my way to teach myself most of course because my maths teachers just tell everybody to memorise everything rather than understand it

Doesn't particularly matter what side the numbers/coefficients go on I just think it's better to keep the Xs on the right hand side and the coeffs on the left with the n choose r
Yes. Recall you can rewrite x^a . x^b as x^(a+b)

Who else is absoluteley done for hahaha, im petrified for Thursday