2. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of (3+bx)^5 where $b$ is a non-zero constant - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 2

To find the first three terms of the binomial expansion $(3 + bx)^5$, we use the Binomial Theorem, which states that:

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

Here, let $a = 3$, $b = bx$, and $n = 5$.

First Term ($k=0$):

When $k=0$:

$T_0 = {5 \choose 0}(3)^{5}(bx)^{0} = 1 \cdot 243 \cdot 1 = 243$

Second Term ($k=1$):

When $k=1$:

$T_1 = {5 \choose 1}(3)^{4}(bx)^{1} = 5 \cdot 81 \cdot bx = 405bx$

Third Term ($k=2$):

When $k=2$:

$T_2 = {5 \choose 2}(3)^{3}(bx)^{2} = 10 \cdot 27 \cdot (bx)^{2} = 270b^2x^2$

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

$243 + 405bx + 270b^2x^2$