| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2002 |
| Session | Specimen |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Arithmetic/geometric progression coefficients |
| Difficulty | Hard +2.3 This AEA question requires multiple sophisticated steps: deriving a relationship from GP conditions on binomial coefficients, solving a Diophantine-style problem to find integer pairs, numerical solving for n given k, and geometric series estimation with logarithms. The combination of algebraic manipulation, number-theoretic reasoning, and multi-part integration exceeds typical A-level significantly. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
6.Given that the coefficients of $x , x ^ { 2 }$ and $x ^ { 4 }$ in the expansion of $( 1 + k x ) ^ { n }$ ,where $n \geq 4$ and $k$ is a positive constant,are the consecutive terms of a geometric series,
\begin{enumerate}[label=(\alph*)]
\item show that $k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }$ .
\item Given further that both $n$ and $k$ are positive integers,find all possible pairs of values for $n$ and $k$ .You should show clearly how you know that you have found all possible pairs of values.
\item For the case where $k = 1.4$ ,find the value of the positive integer $n$ .
\item Given that $k = 1.4 , n$ is a positive integer and that the first term of the geometric series is the coefficient of $x$ ,estimate how many terms are required for the sum of the geometric series to exceed $1.12 \times 10 ^ { 12 }$ .[You may assume that $\log _ { 10 } 4 \approx 0.6$ and $\log _ { 10 } 5 \approx 0.7$ .]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2002 Q6 [18]}}