4. (a) Find the first three terms, in ascending powers of $x$, of the binomial expansion of \[ \frac{1}{\sqrt{4 - x}} \] giving each coefficient in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 2

To expand $\frac{1}{\sqrt{4 - x}}$, we can rewrite it as:

$\frac{1}{(4 - x)^{1/2}}$

Using the binomial expansion for $(1 + u)^{n}$, where $u = -\frac{x}{4}$ and $n = -\frac{1}{2}$:

$(1 + u)^{n} = 1 + nu + \frac{n(n-1)}{2!}u^2 + \cdots$

We adjust this to account for our specific values:

1. First Term: $1$

2. Second Term: $-\frac{1}{2} \left(-\frac{x}{4}\right) = \frac{x}{8}$

3. Third Term: $\frac{-\frac{1}{2} \left(-\frac{1}{2}\right)}{2!} \left(-\frac{x}{4}\right)^2 = \frac{3x^2}{128}$

Thus, the first three terms are:

$1 + \frac{x}{8} + \frac{3x^2}{128}$