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

To find the first three terms of the binomial expansion of $(1 + px)^p$, we use the Binomial Theorem:

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

In our case:

• Let $a = 1$
• Let $b = px$
• Let $n = p$

The first three terms are:

1. When $k=0$:
   ${p \choose 0}(1)^{p}(px)^0 = 1$
2. When $k=1$:
   ${p \choose 1}(1)^{p-1}(px)^1 = p(px) = p(px) = p^1x = p px$
3. When $k=2$:
   ${p \choose 2}(1)^{p-2}(px)^2 = {p(p-1) \over 2!}(p^2x^2) = {p(p-1) \over 2}p^2x^2$

So, the first three terms are:

• $1$
• $p px$
• $\frac{p(p-1)}{2}p^2 x^2$ [Let:q = \frac{p(p-1)}{2}p^2]