maths question

hi, please could i have help on this question? I'm unsure how to answer 3 aii- https://www.ocr.org.uk/Images/643528-question-paper-pure-mathematics-and-statistics.pdf
I've tried using the general formula but im not sure how they got (n+1)x^n?
thanks!

You could pretty much guess it from ai) and its a "write down" so no working is required. But what are you not sure about/what did you do?

for ai i got what is given on the ms. then i tried using the formula in the formula booklet and did -2(-1-n)/n! x^n and simplified to get 2/n x^n which was incorrect?

for ai i got what is given on the ms. then i tried using the formula in the formula booklet and did -2(-1-n)/n! x^n and simplified to get 2/n x^n which was incorrect?

(a)(i): (1-x)⁻² ≈ 1(x⁰) + 2x + 3x² + 4x³
It's a 1 mark question and a beginning question, it is meant to not be too complicated.
We just use pattern spotting to conclude that the general term is (n+1)xⁿ

Its a 1 marker and the question says write down, so no working should be required. Also your general term is different from the terms in part ai) so it cant be right (n=1,2,3).
The binomial series in the qp is for (1+x)^n and the rth term is
n(n-1)...(n-r+1) / r! x^r
Here n=-2, x=-x and r=n (abusing notation somewhat). Can you get it / spot it now?

Sorry, I’ve spotted the pattern mentioned below, is that what you were getting at?

Sure. They ask you to write down 4 terms in ai) and its aii) a follow on part as mentioned in #1.
You could also get it from the general formula as for instance, the 5th term would be
(-2)(-3)(-4)(-5) / 4! (-x)^4
and you can easily see the 4! cancels leaving 5 on the numerator and the negative signs cancel so 5x^4 and its a pattern that obviously generalises. Its more work than needed for this 1 mark/write down question part, though you should be able to do it for practice.