Binomial - Find values of a, b and n

Question

I'm very confused. Would we have to expand the brackets?

If the first three terms of the expansion $(a-2x)^n$ are $1-16x+bx^2 - ...$ then find the values of $a$, $b$ and $n$?

Thank you so much!

Answer 1

From the looks of it, it seems like the question asks for knowledge of the binomial theorem. It goes like this:

$ (x+y)^n = \sum_{k=0}^n {n \choose k} x^{n - k} y^k $

Now, you can use this to calculate for what $k$ and $n$ you get the right factor at $-16x$. Something we are able to deduce is that $a=1$. Seeing as the first term is $1$.

An interesting property of binomials is that - when raised to the $n$th power - the coefficients can be found in Pascal's triangle:

By using this fact, you'll easily be able to find the right value for $n$, and with it $b$.

Good luck!

Answer 2

As said, this is asking you to use the binomial expansion formulae. If you consider the lower order terms in $x$, you will show that $$(a-2x)^n=a^n-2 n a^{n-1}x+2n (n-1) a^{n-2} x^2+\cdots$$ which has to be equal to $1-16x+bx^2+\cdots$

Then, $????$

I am sure that you can take it from here.