| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 7 |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficients in arithmetic/geometric progression |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question requiring standard techniques: writing out terms with binomial coefficients, setting up an arithmetic sequence condition, and solving a cubic that factors nicely. The algebra is routine and the cubic conveniently factors with n as a common factor, making it easier than average but not trivial due to the multi-step nature and 7 total marks. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04h Arithmetic sequences: nth term and sum formulae |
Consider the binomial expansion of $\left(1 + \frac{x}{5}\right)^n$ in ascending powers of $x$, where $n$ is a positive integer.
\begin{enumerate}[label=(\roman*)]
\item Write down the first four terms of the expansion, giving the coefficients as polynomials in $n$. [1]
\end{enumerate}
The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show that $n^3 - 33n^2 + 182n = 0$. [3]
\item Hence find the possible values of $n$ and the corresponding values of the common difference. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q12 [7]}}