Maths AQA C4 Help - Binomial Expansions

a) Determine the binomial expansion of (1-X/10)^-3 in ascending powers of X, up to and including the term in X^3.

b) Show that the coefficients of X^n in this expansion is K(n+1)(n+2)*1/10^n for a rational number K whose value is to be determined.

I can do part a, where I got 1 + X/30 + (3X^2)/25 + (x^3)/100. However my textbook offers no help with regards to part b, any ideas?😕

It's more of an extension question designed to be a bit harder I think. Write out maybe the first four or five terms and don't simplify the coefficients and try and find the general pattern which will simplify to the answer.

What would the first terms be though?

Use the binomial theorem.

So how do I do that if it's not in the form (a+bX)^n ?

Somebody please start this for me, I beg. I've been stuck on this for hours🙏🏻

You expanded wrong for one of them

Variable : Coefficient

$x^0 : 1$

$x^1 : 3 \cdot 10^{-1}$

$x^2 : 6 \cdot 10^{-2}$

$x^3 : 10 \cdot 10^{-3}$

$x^4 : 15 \cdot 10^{-4}$


Notice any patterns? What types of numbers are 1, 3, 6, 10, 15, ...?

Thank you for your help but I don't really see what you're indicating😪

You need to find a relationship between the exponent of $x$ and its coefficient.

So essentially, if you were to carry on with the table above, how would it look like for the general term $x^n$??

Clearly you can notice that at first, there is a $10^{-n}$ involved as a multiple of $x^n$ so carrying the table on would give

$x^ n : K\cdot 10^{-n}$

and now you just need to express $K$ in terms of $n$

So can you some sort of relationship going on here? I gave you a hint for the numbers $1,3,6,10,15$ which are part of family of numbers which have a formula that you can use.

I understand all of that now😄 I see that with the numbers you add on an extra one each time but I don't see how you use it in terms of n. Wouldn't you be able to carry it on forever? Why would it stop at (n+2)? Why not (n+3) etc?