| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expansion of (a+bx^m)^n |
| Difficulty | Hard +2.3 This AEA question requires sophisticated manipulation of the binomial series across multiple connected parts: deriving a non-standard form involving central binomial coefficients, algebraic substitution, differentiation of series, and finally evaluating a specific numerical sum. The multi-step reasoning, need to recognize differentiation as the key technique in part (c), and the extended chain of dependencies make this substantially harder than typical A-level questions, though the individual techniques are within the A-level syllabus. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<14.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
Given that
$$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$
(a) show that
$$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$
[5]
(b) show that $(9 - 4x^2)^{-\frac{1}{2}}$ can be written in the form $\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}$ and give $q$ in terms of $r$.
[3]
(c) Find $\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}$
[3]
(d) Hence find the exact value of
$$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$
giving your answer as a rational number.
[2]
\hfill \mbox{\textit{Edexcel AEA 2014 Q4 [13]}}