| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2004 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Series expansion of rational function |
| Difficulty | Challenging +1.3 This is a structured multi-part question that guides students through binomial series manipulation and summation. Part (a) is routine binomial expansion, parts (b-c) require recognizing series relationships but with clear hints ('hence'), and part (d) applies the formula with straightforward substitution. While it requires multiple techniques and careful algebraic manipulation across several steps, the scaffolding significantly reduces the problem-solving demand compared to questions requiring independent insight. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04g Sigma notation: for sums of series1.04i Geometric sequences: nth term and finite series sum |
\begin{enumerate}[label=(\alph*)]
\item For the binomial expansion of $\frac{1}{(1-x)^2}$, $|x| < 1$, in ascending powers of $x$,
\begin{enumerate}[label=(\roman*)]
\item find the first four terms,
\item write down the coefficient of $x^n$. [2]
\end{enumerate}
\item Hence, show that, for $|x| < 1$, $\sum_{n=1}^{\infty} nx^n = \frac{x}{(1-x)^2}$. [2]
\item Prove that, for $|x| < 1$, $\sum_{n=1}^{\infty} (an+1)x^n = \frac{(a+1)x-x^2}{(1-x)^2}$, where $a$ is a constant. [4]
\item Hence evaluate $\sum_{n=1}^{\infty} \frac{5n+1}{2^{3n}}$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2004 Q2 [10]}}