(a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (1 + px)^n, where p is a constant - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 2

To find the first three terms in the expansion of

$(1 + px)^n$,

we'll apply the binomial theorem, which states:

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

In this case, let:

• $a = 1$
• $b = px$

Thus,

$(1 + px)^n = \sum_{k=0}^{n} \binom{n}{k} (1)^{n-k} (px)^k$

The first three terms correspond to $k = 0, 1, 2$:

1. For $k = 0$:
   • $\binom{n}{0}(1)^{n}(px)^0 = 1$
2. For $k = 1$:
   • $\binom{n}{1}(1)^{n-1}(px)^1 = n(px) = pnx$
3. For $k = 2$:
   • $\binom{n}{2}(1)^{n-2}(px)^2 = \frac{n(n-1)}{2}(px)^2 = \frac{n(n-1)p^2x^2}{2}$

Thus, the first 3 terms are:

$1, pnx, \frac{n(n-1)p^2x^2}{2}$.

Given that these terms match with those provided in the question:

• The first term is 1,
• The second term is $36x$, leading to $pn = 36$, and
• The third term is $qx^2$, leading to $\frac{n(n-1)p^2}{2} = q$.