| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.3 This is a structured binomial expansion question with clear scaffolding. Part (a) requires routine application of the binomial theorem for negative indices, part (b) is direct recall of geometric series, and part (c) requires recognizing that the series is the derivative of a geometric series—a standard technique. While it involves multiple steps, each is well-signposted and uses familiar A-level methods, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04j Sum to infinity: convergent geometric series |r|<1 |
In this question you should assume that $-1 < x < 1$.
\begin{enumerate}[label=(\alph*)]
\item For the binomial expansion of $(1 - x)^{-2}$
\begin{enumerate}[label=(\roman*)]
\item find and simplify the first four terms, [2]
\item write down the term in $x^n$. [1]
\end{enumerate}
\item Write down the sum to infinity of the series $1 + x + x^2 + x^3 + \ldots$. [1]
\item Hence or otherwise find and simplify an expression for $2 + 3x + 4x^2 + 5x^3 + \ldots$ in the form $\frac{a - x}{(b - x)^2}$ where $a$ and $b$ are constants to be determined. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2020 Q3 [7]}}