| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2019 |
| Marks | 10 |
| Topic | Binomial Theorem (positive integer n) |
| Type | Coefficients in arithmetic/geometric progression |
| Difficulty | Challenging +1.2 This is a multi-part binomial expansion problem requiring algebraic manipulation to establish a relationship between coefficients forming a GP, then applying this in specific cases. Part (a) involves setting up equations from the GP condition and algebraic simplification (standard but multi-step). Parts (b) and (c) are computational applications. The question requires careful algebra and understanding of GP properties, but follows a clear path without requiring novel insight—moderately above average difficulty for Further Maths. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
In the question you must show detailed reasoning
Given that the coefficients of $x$, $x^2$ and $x^4$ in the expansion of $(1 + kx)^n$ are the consecutive terms of a geometric series, where $n \geq 4$ and $k$ is a positive constant
\begin{enumerate}[label=(\alph*)]
\item Show that $$k = \frac{6(n-1)}{(n-2)(n-3)}$$ [4]
\item For the case when $k = \frac{7}{2}$, find the value of $n$. [2]
\item Given that $k = \frac{7}{5}$, $n$ is a positive integer, and that the first term of the geometric series is the coefficient of $x$, find the number of terms required for the sum of the geometric series to exceed $1.12 \times 10^{12}$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2019 Q11 [10]}}