| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown power and constant |
| Difficulty | Standard +0.3 This is a standard binomial expansion question requiring students to equate coefficients to find unknown constants. While it involves multiple steps (finding a and n from two equations, then calculating k), the method is routine and commonly practiced in C4. The algebra is straightforward once the binomial coefficient formulas are applied. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((1 + ax)^n = 1 + nax + \frac{n(n-1)}{2}(ax)^2 + \ldots\) | B1 | |
| \(\therefore an = -4, \quad \frac{a^2n(n-1)}{2} = 24\) | B1 | |
| \(\Rightarrow a = \frac{-4}{n}, \text{ sub. } \Rightarrow \frac{16}{n^2} \cdot \frac{n(n-1)}{2} = 24\) | M1 A1 | |
| \(8(n-1) = 24n, \quad n = -\frac{1}{2}, a = 8\) | M1 A1 | |
| (b) \((1 + 8x)^{-\frac{1}{2}} = \ldots + \frac{(-1)(-\frac{3}{2})(-\frac{5}{2})}{3 \times 2}(8x)^3 + \ldots\) | M1 | |
| \(\therefore k = -\frac{5}{16} \times 512 = -160\) | A1 | (8 marks) |
**(a)** $(1 + ax)^n = 1 + nax + \frac{n(n-1)}{2}(ax)^2 + \ldots$ | B1 |
$\therefore an = -4, \quad \frac{a^2n(n-1)}{2} = 24$ | B1 |
$\Rightarrow a = \frac{-4}{n}, \text{ sub. } \Rightarrow \frac{16}{n^2} \cdot \frac{n(n-1)}{2} = 24$ | M1 A1 |
$8(n-1) = 24n, \quad n = -\frac{1}{2}, a = 8$ | M1 A1 |
**(b)** $(1 + 8x)^{-\frac{1}{2}} = \ldots + \frac{(-1)(-\frac{3}{2})(-\frac{5}{2})}{3 \times 2}(8x)^3 + \ldots$ | M1 |
$\therefore k = -\frac{5}{16} \times 512 = -160$ | A1 | (8 marks)\begin{enumerate}
\item The first four terms in the series expansion of $( 1 + a x ) ^ { n }$ in ascending powers of $x$ are
\end{enumerate}
$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$
where $a , n$ and $k$ are constants and $| a x | < 1$.\\
(a) Find the values of $a$ and $n$.\\
(b) Show that $k = - 160$.\\
3. continued\\
\hfill \mbox{\textit{Edexcel C4 Q3 [8]}}