2. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $(2 + kx)^7$ where $k$ is a non-zero constant - Edexcel - A-Level Maths Pure - Question 4 - 2018 - Paper 4

To find the first four terms of the binomial expansion of $(2 + kx)^7$, we can use the binomial theorem, which states:

$(a + b)^n = \sum_{r=0}^{n} {n \choose r} a^{n-r} b^r$

For our expression, let $a = 2$, $b = kx$, and $n = 7$. Then the first four terms are:

1. For $r=0$: ${7 \choose 0} (2)^7 (kx)^0 = 1 \cdot 128 \cdot 1 = 128$

2. For $r=1$: ${7 \choose 1} (2)^6 (kx)^1 = 7 \cdot 64 \cdot kx = 448kx$

3. For $r=2$: ${7 \choose 2} (2)^5 (kx)^2 = 21 \cdot 32 \cdot k^2x^2 = 672k^2x^2$

4. For $r=3$: ${7 \choose 3} (2)^4 (kx)^3 = 35 \cdot 16 \cdot k^3x^3 = 560k^3x^3$

Thus, the first four terms in ascending powers of $x$ are:

$128 + 448kx + 672k^2x^2 + 560k^3x^3$