| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove summation with exponentials |
| Difficulty | Challenging +1.2 This is a structured induction proof with exponentials that requires careful algebraic manipulation. Part (a) provides scaffolding for the inductive step, and while the algebra is somewhat involved (combining terms with powers of 1/2), the question follows a standard induction template with clear guidance. It's harder than a basic induction proof due to the exponential terms and multi-step algebra, but remains a typical FP2 examination question without requiring novel insight. |
| Spec | 4.01a Mathematical induction: construct proofs |
(a) Express $(k+1)^2 + 5(k+1) + 8$ in the form $k^2 + ak + b$, where $a$ and $b$ are constants.
[1 mark]
(b) Prove by induction that, for all integers $n \geq 1$,
$$\sum_{r=1}^{n} r(r+1) \left(\frac{1}{2}\right)^{r-1} = 16 - \left(\frac{1}{2}\right)^{n-1}(n^2 + 5n + 8)$$
[6 marks]3
\begin{enumerate}[label=(\alph*)]
\item Express $( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8$ in the form $k ^ { 2 } + a k + b$, where $a$ and $b$ are constants.
\item Prove by induction that, for all integers $n \geqslant 1$,
$$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2014 Q3 [7]}}